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The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 25
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 66718)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 25
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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text
string | meta
dict |
|---|---|
\section{Introduction}
Harmonic maps from a Riemann surface into a Lie group $\mathrm{G}$, with Lie algebra $\mathfrak{g}$, correspond to certain holomorphic maps,
the \emph{extended solutions}, into the group $\Omega \mathrm{G}$
of based smooth loops in $\mathrm{G}$ \cite{uhlenbeck_1989}.
If the Fourier series associated to an extended solution $\Phi$ has finitely many terms, we say that $\Phi$ and the corresponding harmonic map have \emph{finite uniton number}. It is well known that all harmonic maps from the two-sphere have finite uniton number \cite{uhlenbeck_1989}.
When $\mathrm{G}$ has trivial center, Burstall and Guest \cite{burstall_guest_1997} have classified harmonic maps with finite uniton number from a Riemann surface $M$ into $\mathrm{G}$ in terms of the pieces of the Bruhat decomposition of
$$\Omega_\mathrm{alg}\mathrm{G}=\{\gamma\in\Omega \mathrm{G} \,|\,\mbox{$\gamma$ and $\gamma^{-1}$ have finite Fourier series}\}.$$
More precisely, each piece of the Bruhat decomposition coincides with the unstable manifold associated to the flow of the gradient vector field of a certain Morse-Bott function defined on the K\"{a}hler manifold $\Omega_\mathrm{alg}\mathrm{G}$; these unstable manifolds are parameterized by the elements of a certain integer lattice $\mathfrak{I}(\mathrm{G})$ in $\g$; any extended solution with finite uniton number takes values, off a discrete subset of $M$, in one of these unstable manifolds, and so corresponds to some element $\xi\in \mathfrak{I}(\mathrm{G})$; when $G$ has trivial center and maximal torus with dimension $n$, there is a finite subset $\Xi_\mathrm{can}(G)$ of the integer lattice $\mathfrak{I}(\mathrm{G})$ with $2^n$ elements so that any harmonic map from $M$ to $G$ corresponds to an extended solution with values, off a discrete subset, on the unstable manifold associated to some \emph{canonical element} $\xi\in \Xi_\mathrm{can}(G)$. Among such extended solutions, a distinguished type is that of $\mathrm{S}^1$-\emph{invariant} extended solutions, which correspond to harmonic maps admitting \emph{super-horizontal} holomorphic lifts into a certain \emph{twistor space}. For example, all harmonic spheres in $\mathrm{S}^n$ and $\C \mathrm{P}^n$ arise in this way (see \cite{eells_lemaire} and references therein).
In the present paper, we classify all harmonic maps with finite uniton number from $M$ into the special unitary group $\mathrm{SU}(n)$ and corresponding inner symmetric spaces, the Grassmannians $\mathrm{Gr}(k,n)$ of $k$-dimensional subspaces of $\C^n$, in terms of certain pieces of the Bruhat decomposition of $\Omega_{\mathrm{alg}} \SU(n)$.
For that we use the results of \cite{correia_pacheco_3} in order to generalize the notion of canonical element of $\mathfrak{I}(\mathrm{G})$ to the case where $G$ has not necessarily trivial center (recall that the center of $\SU(n)$ is isomorphic to the cyclic group $\mathbb{Z}_n$). Moreover, in the setting of the Grassmannian model for loops groups \cite{pressley_segal} we give a description of the ``Frenet frame data" for such harmonic maps in a given class. The Grassmannian model for loop groups was exploited for the first time in the study of harmonic maps into the unitary group $\mathrm{U}(n)$ by Segal \cite{segal_1989}. More recently, the Grassmannian model has been used in the study of harmonic maps into other Lie groups and their inner symmetric spaces \cite{correia_pacheco_3,pacheco_2006,svensson_wood_2010}.
We remark that Ferreira, Sim\~{o}es and Wood \cite{ferreira_simoes_wood_2010} established an algebraic formula for all harmonic
maps with finite uniton number from a Riemann surface $M$ into the unitary
group $\mathrm{U}(n)$ in terms of freely chosen meromorphic functions on $M$ and their derivatives. Since any such harmonic map has constant determinant, this formula can be easily applied in order to obtain all
harmonic maps with finite uniton number from $M$ into $\mathrm{SU}(n)$. However, it does not clarifies how to choose the meromorphic functions in order to produce harmonic maps associated to extended solutions in the class of a given element $\xi\in \Xi_\mathrm{can}(\mathrm{SU}(n))$.
In this paper we shall see how to do that in the case of harmonic maps associated to $\mathrm{S}^1$-invariant extended solutions.
\section{Grassmannian model for loop groups}
Let us start by recalling from Pressley and Segal \cite{pressley_segal} some standard definitions and facts concerning the Grassmannian model for loop groups.
Fix on $\mathbb{C}^{n}$ the standard complex inner product $\langle
\cdot,\cdot\rangle$ and let $e_{1},\ldots,e_{n}$ be the standard basis vectors for
$\mathbb{C}^{n}$. Given a complex subspace $V\subset \C^n$, we denote by $\pi_V$ the orthogonal projection onto $V$.
Let $H^n$ be the Hilbert space
of square-summable $\C^{n}$-valued
functions on the
circle and $ \langle\cdot,\cdot\rangle_H$ the induced complex inner product. This is the closed space generated by
the functions
$\lambda\mapsto\lambda^{i}e_{j}$, with $i\in\mathbb{Z}$ and $j=1,\ldots,n$.
Consider the closed subspace $H_+^n$ of $H^n$ defined by
$H_+^n=\mbox{Span}\{ \lambda^{i}e_{j}\,|\,i\geq 0,\,
j=1,\ldots,n\}.$
Let $\mbox{\emph{Grass}}(H^n)$ denote the set of all closed vector
subspaces $W\subset H^n$ such that:
the projection map $W\rightarrow H_+^n$ is
Fredholm, and the projection map $W\rightarrow
{H_+^n}^\perp$ is Hilbert-Schmidt;
the images of the projection maps $W^{\perp}\rightarrow H_+^n$, $W\rightarrow
{H_+^n}^\perp$ are contained in $C^{\infty}(\mathrm{S}^{1};\mathbb{C}^{n})$. Define
$$\mathrm{Gr}^n=\{W\in \mbox{\emph{Grass}}(H^n)\,|\, \lambda W\subseteq
W \}.$$
The action of the infinite-dimensional Lie group
$\Lambda \mathrm{U}(n)=\big\{\gamma:S^1\to \mathrm{U}(n)\,|\, \mbox{$\gamma$ is smooth}\big\}$
on $\mathrm{Gr}^n$ defined by $\gamma W=\{\gamma f\,|\,f\in W\}$ is transitive. By considering Fourier series, it is easy to see that the
isotropy subgroup at $H_+^n$ is precisely
${\mathrm{U}}(n)$. Hence
$\mathrm{Gr}^n \cong \Lambda \mathrm{U}(n)/\mathrm{U}(n)\cong \Omega \mathrm{U}(n).$ This homogeneous space carries a natural invariant structure of K\"{a}hler manifold.
\begin{rem}\label{rem} Given $W\in \mathrm{Gr}^n$, then
$\dim W \ominus \lambda W=n$, where $W \ominus
\lambda W$ denotes the orthogonal complement of $\lambda W$ in
$W$, and the evaluation map $\rm{ev}_\lambda:W\ominus \lambda W:\to \C^n$ at $\lambda\in\mathrm{S}^1$ is a unitary isomorphism. If we choose an orthonormal basis for $W\ominus\lambda W$,
$\{w_1,\ldots,w_n\}$, we can put the vector-valued functions
$w_i$ side by side to form an $(n\times n)$-matrix valued
function $\gamma$ on $\mathrm{S}^1$, that is, a loop $\gamma\in\Lambda
{\mathrm{U}}(n)$. It can be shown that $W=\gamma H^n_+$.
\end{rem}
A loop
$\gamma \in\Omega \mathrm{U}(n)$ is said to be \emph{algebraic} if both $\gamma$ and $\gamma^{-1}$ have finite
Fourier series. Denote by
$\Omega_{\rm{alg}}\mathrm{U}(n)$ the subgroup of algebraic loops. This
subgroup acts on
$$\mathrm{Gr}^n_{\mathrm{alg}}=\{W \in \mathrm{Gr}^n\,|\,
\lambda^k H_+^n \subseteq W\subseteq \lambda^{-k}
H_+^n\,\,\textrm{for some } k\in\mathbb{N} \},
$$
and we have $\mathrm{Gr}^n_{\mathrm{alg}}\cong
\Omega_{\mathrm{alg}}{\mathrm{U}}(n)$. Given $r\leq k$, we set $$\Omega_r^k\mathrm{U}(n)=\big\{\gamma\in \Omega_{\mathrm{alg}} \mathrm{U}(n)\,|\,\gamma(\lambda)=\sum_{i=r}^{k}\gamma_i\lambda^i\big\},$$ where the coefficients $\gamma_i$ are constant $(n\times n)$-complex matrices.
If $\mathrm{G}$ is a subgroup of $\mathrm{U}(n)$, we shall denote by $\mathrm{Gr}(\mathrm{G})$ the subspace of $\mathrm{Gr}^n$ that corresponds to $\Omega \mathrm{G}$ and by ${\mathrm{Gr}}_{\mathrm{alg}}(\mathrm{G})$ the subspace of $\mathrm{Gr}(\mathrm{G})$ that corresponds to $\Omega_{\mathrm{alg}}\mathrm{G}$.
\section{The Bruhat Decomposition of $\mathrm{Gr}_{\mathrm{alg}}(\mathrm{G})$}
Next we describe the Bruhat decomposition for algebraic loop groups. For more details we refer the reader to \cite{burstall_guest_1997} and \cite{pressley_segal}.
Consider a compact matrix semi-simple Lie group $\mathrm{G}$. Fix a maximal torus $\mathrm{T}$ of $\mathrm{G}$ with Lie algebra $\mathfrak{t}\subset \mathfrak{g}$, let $\Delta\subset \sqrt{-1} \mathfrak{t}^*$ be the corresponding
set of roots and denote by $\g_\alpha$ the root space of $\alpha\in \Delta$. The integer lattice $\mathfrak{I}(\mathrm{G}) =
(2\pi)^{-1} \exp^{-1}(e)\cap \mathfrak{t}$ may be identified with the group of homomorphisms $\mathrm{S}^1\to \mathrm{T}$, by
associating to $\xi\in \mathfrak{I}(\mathrm{G})$ the homomorphism
$\gamma_\xi$ defined by $\gamma_\xi(\lambda)=\exp{(-\sqrt{-1}\ln(\lambda)\xi)}$.
Let $H_1,\ldots,H_k\in \mathfrak{t}$ be dual to the positive simple roots $\alpha_1,\ldots,\alpha_k\in \Delta^+$ of $\g^\C$:
$\alpha_i(H_j)=\sqrt{-1}\,\delta_{ij}$. By applying the well-known formula $\mathrm{Ad}(\exp(\eta))=\exp(\mathrm{ad}(\eta))$, for all $\eta\in\g^\C$, we can easily check that
the integer lattice $\mathfrak{I}(G)$ is contained in $\mathbb{Z}H_1\oplus\ldots\oplus \mathbb{Z}H_k$.
Denote by $\g^\xi_i$ the $\sqrt{-1}\,i$-eigenspace of $\mathrm{ad}{\xi}$, with $i\in\mathbb{Z}$.
We have on $\mathfrak{g}^\C$ the structure of graded Lie algebra:
\begin{equation*}
\g^\C=\!\!\!\bigoplus_{i\in\{-r(\xi)\ldots,r(\xi)\}}\!\!\!\mathfrak{g}^\xi_i,\quad [\mathfrak{g}^\xi_i, \mathfrak{g}^\xi_j]\subset \mathfrak{g}^\xi_{i+j},
\end{equation*}
where $r(\xi)=\mathrm{max}\{i\,|\,\,\g_i^\xi\neq 0\}$, and
\begin{equation}\label{gis}
\g_i^\xi=\!\!\bigoplus_{\alpha(\xi)=\sqrt{-1}\,i}\!\!\g_\alpha.
\end{equation}
Set $\Lambda^+\mathrm{G}^\C=\{\gamma:S^1\to \mathrm{G}^\C\,|\,\,\mbox{$\gamma$ extends holomorphically for $|\lambda|\leq 1$}\}$.
For each
$\xi\in \mathfrak{I}(\mathrm{G})$, we write
$\Omega_\xi=\{g\gamma_\xi g^{-1} \,|\,\, g \in \mathrm{G}\},$ the conjugacy class of homomorphisms $\mathrm{S}^1\to \mathrm{G}$ which contains $\gamma_\xi$.
This is a complex homogeneous space:
$$\Omega_\xi\cong \mathrm{G}^\C\big/\mathrm{P}_\xi,\,\mbox{with}\,\mathrm{P}_\xi=\mathrm{G}^\C\cap \gamma_\xi\Lambda^+\mathrm{G}^\C\gamma_\xi^{-1}.$$
Taking account that
$\gamma_\xi X_j\gamma_\xi^{-1}=\lambda^jX_j$ for each $X_j\in \g^\xi_j$ (this is a direct consequence of the formula $\mathrm{Ad}(\exp(\eta))=\exp(\mathrm{ad}(\eta))$, for all $\eta\in\g^\C$), one can easily check that the Lie algebra of the isotropy subgroup $\mathrm{P}_\xi$ is precisely the parabolic subalgebra $\mathfrak{p}_\xi=\bigoplus_{i\leq 0}\g^\xi_i$ induced by $\xi$.
Now, fix a positive set of roots $\Delta^+\subset \Delta$ and set $\mathfrak{I}'(\mathrm{G})=\{\xi\in\mathfrak{I}(\mathrm{G})|\, \mbox{$\alpha(\xi)\geq 0$ for all $\alpha\in \Delta^+$}\}.$ We have:
\begin{thm}\cite{pressley_segal}
\emph{Bruhat decomposition: }$\mathrm{Gr}_\mathrm{alg}(\mathrm{G})=\bigcup_{\xi\in \mathfrak{I}'(\mathrm{G})}\Lambda^+_\mathrm{alg}\mathrm{G}^\C\gamma_\xi H_+^n$.
\end{thm}
Define $U_\xi(\mathrm{G})\subset \Omega_{\mathrm{alg}}\mathrm{G}$ by $U_\xi(\mathrm{G}) H_+^n=\Lambda^+_{\mathrm{alg}}\mathrm{G}^\C\gamma_\xi H^n_+.$ This is also a complex homogeneous space
of $\Lambda^+_{\mathrm{alg}}\mathrm{G}^\C$
with isotropy subgroup at $\gamma_\xi$ given by $\Lambda^+_{\mathrm{alg}}\mathrm{G}^\C\cap \gamma_\xi \Lambda^+\mathrm{G}^\C \gamma_\xi^{-1}.$ Moreover, $U_\xi(\mathrm{G})$ carries the structure of holomorphic vector bundle over $\Omega_\xi$ whose bundle map
$u_\xi:U_\xi(\mathrm{G})\to \Omega_\xi$ is precisely the natural map
$[\gamma]\mapsto [\gamma(0)]$. The holomorphic tangent bundle of $U_\xi(\mathrm{G})$ is given by
\begin{equation}\label{holm}
T^{1,0}U_\xi(\mathrm{G})\cong \Lambda^+_{\mathrm{alg}}\mathrm{G}^\C\times_{\Lambda^+_{\mathrm{alg}}\mathrm{G}^\C\cap \gamma_\xi\Lambda^+\mathrm{G}^\C\gamma_\xi^{-1}}\Lambda^+_{\mathrm{alg}}\mathfrak{g}^\C\big/\Lambda^+_{\mathrm{alg}}\mathfrak{g}^\C\cap \gamma_\xi\Lambda^+\mathfrak{g}^\C\gamma_\xi^{-1}
\end{equation}
In terms of the Grassmannian model, the bundle map $u_\xi:U_\xi(\mathrm{G})\to\Omega_\xi$ can be described as follows.
Take $\gamma\in U_\xi(\mathrm{G})$ and $W=\gamma H^n_+\in \mathrm{Gr}_{\mathrm{alg}}(\mathrm{G})$, with
$\lambda^rH^n_+\subset W\subset\lambda^{-s}H^n_+$.
Fix $\Psi\in \Lambda^+_{\mathrm{alg}}\mathrm{G}^\C$ such that
$W=\Psi\gamma_\xi H^n_+$. Write
\begin{equation*}\label{flag}
\gamma_\xi H^n_+=\lambda^{-s}A^\xi_{-s}+\ldots+\lambda^{r-1}A^\xi_{r-1}+\lambda^rH^n_+,
\end{equation*} where the subspaces $A^\xi_i$ define a flag
\begin{equation}\label{flag1}
\{0\}=A^\xi_{-s-1}\subsetneq A^\xi_{-s}\subseteq A^\xi_{-s+1}\subseteq \ldots\subseteq A^\xi_{r-1}\subsetneq A^\xi_r=\C^n.
\end{equation}
Set $A_i=\Psi(0)A^\xi_i= p_i(W\cap\lambda^iH^n_+),$ where $p_i: H^n \to\C^n$ is defined by
\begin{equation}\label{pi}
p_i(\sum\lambda^ja_j)=a_i.
\end{equation} Then
\begin{equation}\label{popo1}
u_\xi(W)=\lambda^{-s}A_{-s}+\ldots+\lambda^{r-1}A_{r-1}+\lambda^rH^n_+.
\end{equation}
Following \cite{correia_pacheco_3}, consider the partial order $\preceq$ over $\mathfrak{I}'(\mathrm{G})$ defined by:
$\xi\preceq \xi'$ if $\mathfrak{p}^{\xi}_i\subseteq \mathfrak{p}^{\xi'}_i$
for all $i\geq 0$, where $\mathfrak{p}_i^\xi=\bigoplus_{j\leq i}\g_j^\xi$.
Given two elements $\xi,\xi'\in \mathfrak{I}'(\mathrm{G})$ such that $\xi\preceq \xi'$, it can be shown \cite{correia_pacheco_3} that
$$\Lambda^+_{\mathrm{alg}}\mathrm{G}^\C\cap \gamma_\xi \Lambda^+\mathrm{G}^\C \gamma_\xi^{-1}\subseteq \Lambda^+_{\mathrm{alg}}\mathrm{G}^\C\cap \gamma_{\xi'} \Lambda^+\mathrm{G}^\C \gamma_{\xi'}^{-1}.$$
This allows us to define, for $\xi\preceq \xi'$, a $\Lambda^+_{\mathrm{alg}}\mathrm{G}^\C$-invariant fibre bundle morphism
$\mathcal{U}_{\xi,\xi'}:U_\xi\to U_{\xi'}$ by
\begin{equation*}\label{uxi}
\mathcal{U}_{\xi,\xi'}(\Psi\gamma_{\xi}H^n_+)=\Psi\gamma_{\xi'}H^n_+, \quad \Psi\in\Lambda^+_{\mathrm{alg}}\mathrm{G}^\C.\end{equation*}
Since the holomorphic structures on $U_\xi(\mathrm{G})$ and $U_{\xi'}(\mathrm{G})$ are induced by the holomorphic structure on $\Lambda^+_{\mathrm{alg}}\mathrm{G}^\C$, the fibre-bundle morphism $\mathcal{U}_{\xi,\xi'}$ is holomorphic.
\section{Harmonic maps into a Lie group}
\subsection{Extended solutions}
Let $M$ be a Riemann surface and $\varphi:M\rightarrow \mathrm{G}\subseteq \mathrm{U}(n)$ a map into
a compact matrix Lie group. Equip $\mathrm{G}$ with a bi-invariant metric.
Define $\alpha=\varphi^{-1}{d}\varphi$ and let $\alpha=\alpha'+\alpha''$
be the type decomposition of $\alpha$ into $(1,0)$ and
$(0,1)$-forms. It is well known \cite{uhlenbeck_1989}
that $\varphi:M\rightarrow \mathrm{G} $ is harmonic if and only if the loop of
$1$-forms given by
\begin{equation}
\label{flcon}
\alpha_\lambda=\frac{1-\lambda^{-1}}{2}
\alpha'+\frac{1-\lambda}{2} \alpha''
\end{equation}
satisfies the Maurer-Cartan equation ${d}\alpha_\lambda + \frac{1}{2}[\alpha_\lambda\wedge \alpha_\lambda]=0$
for each $\lambda\in \mathrm{S}^1$.
Then, if $M$ is simply connected and $\varphi$ is harmonic, we can
integrate to obtain a map $\Phi:M\rightarrow \Omega \mathrm{G}$, the \textit{extended solution} associated to $\varphi$, such that
$\alpha_\lambda=\Phi_\lambda^{-1}{d}\Phi_\lambda$ and $\Phi_{-1}=\varphi$. Conversely, if $\Phi:M\rightarrow \Omega \mathrm{G}$ is an extended solution, that is if it integrates a loop of $1$-form of the form \eqref{flcon}, then $\varphi=\Phi_{-1}:M\rightarrow \mathrm{G}$ is harmonic.
\begin{thm}\cite{burstall_guest_1997}\label{usd}
\emph{Let $\Phi:M\to \Omega_{\mathrm{alg}}\mathrm{G}$ be an extended solution. Then there exists some $\xi\in \mathfrak{I}'(\mathrm{G})$, and some discrete subset $D$ of $M$, such that $\Phi(M\setminus D)\subseteq U_\xi(\mathrm{G})$.}
\end{thm}
Given a smooth map $\Phi:M\setminus D\to U_\xi(\mathrm{G})$, consider $\Psi:M\setminus D \to \Lambda_{\mathrm{alg}}^+\mathrm{G}^\C$ such that $\Phi H^n_+=\Psi\gamma_\xi H^n_+$. Clearly,
$\Psi\gamma_\xi=\Phi b$ for some $b:M\setminus D\to \Lambda^+_{\mathrm{alg}}G^\C.$
Write
\begin{equation*}\label{not}
\Psi^{-1}\Psi_z=\sum_{i\geq 0} X'_i\lambda^i,\,\,\,\,\Psi^{-1}\Psi_{\bar{z}}=\sum_{i\geq 0} X''_i\lambda^i.\end{equation*}
Proposition 4.4 in \cite{burstall_guest_1997} establishes that $\Phi$ is an extended solution if, and only if,
\begin{equation}\label{im}
\mathrm{Im} X'_i\subset \,\mathfrak{p}^\xi_{i+1},\,\,\,\,\mathrm{Im} X''_i\subset \mathfrak{p}^\xi_{i},
\end{equation}
where $\mathfrak{p}_i^\xi=\bigoplus_{j\leq i}\g_j^\xi.$ The second condition says that $\Phi:M\setminus D\to U_\xi(\mathrm{G})$ is holomorphic.
The bundle morphism $\mathcal{U}_{\xi,\xi'}$ and the bundle map $u_\xi$ are well behaved with respect to extended solutions:
\begin{thm}\label{popo}\cite{correia_pacheco_3}
\emph{Given an extended solution $\Phi:M\setminus D\to U_\xi(\mathrm{G})$ and an element $\xi'\in \mathfrak{I}'(\mathrm{G})$ such that $\xi\preceq \xi'$, then
$\mathcal{U}_{\xi,\xi'}(\Phi)=\mathcal{U}_{\xi,\xi'}\circ \Phi:M\setminus D\to U_{\xi'}(\mathrm{G})$ is a new extended solution.}
\end{thm}
\begin{thm}\cite{burstall_guest_1997}
\emph{If $\Phi:M\setminus D\to U_\xi(\mathrm{G})$ is an extended solution, then $u_\xi\circ\Phi:M\setminus D\to \Omega_\xi$ is an extended solution.}
\end{thm}
An \emph{$\mathrm{S}^1$-invariant extended solution} is an extended solution which takes values in $\Omega_\xi$, for some $\xi\in \mathfrak{I}'(\mathrm{G})$.
\subsection{Harmonic maps into inner $\mathrm{G}$-symmetric spaces}
Given a compact (connected) Lie group $\mathrm{G}$,
each connected component of $\sqrt{e}=\{g\in
\mathrm{G}\,|\,\,g^2=e\}$ is a compact inner symmetric space \cite{burstall_guest_1997}. Conversely, any compact (connected) inner $\mathrm{G}$-symmetric space may be immersed in $\mathrm{G}$ as a connected component of $\sqrt{e}$.
Moreover,
the embedding of each component of $\sqrt{e}$ in $\mathrm{G}$ is totally
geodesic. Hence harmonic maps into $\mathrm{G}$-inner symmetric spaces can be viewed as special harmonic maps into $\mathrm{G}$.
As in \cite{burstall_guest_1997}, define the involution $\mathcal{I}:\Omega \mathrm{G} \rightarrow \Omega \mathrm{G}$ by $\mathcal{I}(\gamma)(\lambda)
=\gamma(-\lambda)\gamma(-1)^{-1}.$ Write $$\Omega^\mathcal{I}
\mathrm{G}=\{\gamma\in \Omega \mathrm{G}\,|\,\,\mathcal{I}(\gamma)=\gamma\}$$
for the fixed set of $\mathcal{I}$. Let $M$ be a Riemann surface and
$\Phi:M\rightarrow \Omega^\mathcal{I} \mathrm{G}$ an extended solution. Then
$\varphi=\Phi_{-1}$ defines a harmonic map from $M$ into a
connected component of $\sqrt{e}$.
Conversely, if $\varphi:M\rightarrow\sqrt{e}$ is a harmonic map, there exists an extended
solution $\Phi:M\rightarrow \Omega^\mathcal{I} \mathrm{G}$ such that
$\varphi=\Phi_{-1}$. Under the identification $\Omega \mathrm{G}\cong \mathrm{Gr}(\mathrm{G})$,
$\mathcal{I}$ induces
an involution on $\mathrm{Gr}(\mathrm{G})$, that we shall also denote by $\mathcal{I}$,
and $\Omega^\mathcal{I} \mathrm{G}$ can be identified with
\bdm
\mathrm{Gr}^\mathcal{I}(\mathrm{G})=\{W\in \mathrm{Gr}(\mathrm{G})\,|\,\,
\,\mbox{if $s(\lambda)\in W$ then $s(-\lambda)\in W$}\}.
\edm
Corresponding to the extended solution $\Phi:M\rightarrow \Omega^\mathcal{I}
\mathrm{G}$, consider $W=\Phi H_+:M \rightarrow
\mathrm{Gr}^\mathcal{I}(\mathrm{G})$.
For each $\xi \in \mathfrak{I}'(\mathrm{G})$ we can associate the symmetric space $N_\xi=\{g\gamma_\xi(-1)g^{-1}\,|\,\,g\in G$\}.
If an extended solution takes values in $U_\xi^{\mathcal{I}}(\mathrm{G})=U_\xi(\mathrm{G})\cap \Omega^{\mathcal{I}}\mathrm{G}$, then the corresponding harmonic map takes values in $N_\xi$.
Observe that, for $\xi$ and $\xi'$ in $\mathfrak{I}'(\mathrm{G})$, if $\xi-\xi'\in\mathfrak{I}^{2}(\mathrm{G}):=\pi^{-1}\exp^{-1}(e)\cap \mathfrak{t}$, then $N_\xi=N_{\xi'}$.
Moreover, as shown in \cite{correia_pacheco_3},
if $\xi\preceq \xi'$, then $\mathcal{U}_{\xi,\xi'}(U_\xi^{\mathcal{I}}(\mathrm{G}))\subset U_{\xi'}^{\mathcal{I}}(\mathrm{G})$. To sum up, if we define a new partial order $\preceq_\mathcal{I}$ in $\mathfrak{I}'(\mathrm{G})$ by
\begin{equation*}
\xi\preceq_\mathcal{I}\xi'\,\,\, \mbox{ if}\,\,\, \xi\preceq \xi' \,\,\, \mbox{and}\,\,\, \xi-\xi'\in\mathfrak{I}^{2}(\mathrm{G}),
\end{equation*}
the following holds:
\begin{prop}\label{proposition}
\emph{ If $\xi\preceq_\mathcal{I} \xi'$, then $\mathcal{U}_{\xi,\xi'}(U_\xi^{\mathcal{I}}(\mathrm{G}))\subset U_{\xi'}^{\mathcal{I}}(\mathrm{G})$ and $N_\xi=N_{\xi'}$.}
\end{prop}
\subsection{Extended solutions from the Grassmannian point of view}
Let $W:{M} \rightarrow \mathrm{Gr}(\mathrm{G})$ correspond to a smooth map $\Phi:M\to \Omega \mathrm{G}$ under the
identification $\Omega \mathrm{G} \cong \mathrm{Gr}(\mathrm{G})$, that is $W=\Phi
H^n_+$.
Segal \cite{segal_1989} has observed that $\Phi$ is an extended solution if, and only if, $W$ is a
solution of equations:
\begin{align}
W_z &\subseteq {\lambda}^{-1}W \label{phh},\\ W_{\bar{z}}
& \subseteq W.\label{hh}
\end{align}
Condition \eqref{phh} means that, in any local complex coordinate $z$, $\frac{\partial s}{\partial z}(z)$
is contained in the subspace $\lambda^{-1}W(z)$ of $H^n$, for every
(smooth) map $s : M\rightarrow H^n$ such that $s(z)\in
W(z)$. Inspired by \cite{burstall_rawnsley_1990} (Section F of Chapter 8), we call \eqref{phh} the \emph{pseudo-horizontality} condition. Condition \eqref{hh} is interpreted in a similar way and states that $W$ is a holomorphic vector subbundle of $M\times H^n$.
\begin{rem}
Consider some discrete set $D\subset M$, an element $\xi\in \mathfrak{I}'(\mathrm{G})$ and an extended solution $\Phi:M\setminus D\to U_\xi(\mathrm{G})$. As explained in Remark 2.5 of \cite{correia_pacheco_2}, the bundle
$W=\Phi H^n_+$ can be extended holomorphically to $M$, and, consequently,
$\Phi$ defines a global extended solution from $M$ to $\Omega_{\mathrm{alg}}\mathrm{G}$.
\end{rem}
If $\Phi:M\setminus D\to U_\xi(\mathrm{G})$ is an extended solution and $W=\Phi H^n_+$, then $u_\xi(W)=u_\xi\circ\Phi H_+^n$ is given pointwise by
(\ref{popo1}) and we get holomorphic subbundles $A_i$ of the trivial bundle $\underline{\C}^n=M\times \C^n$ such that
\begin{equation}\label{super}
0\subsetneq A_{-s} \subseteq \ldots \subseteq A_{r-1} \subsetneq A_r=\underline{\C}^n.
\end{equation}
The pseudo-horizontally condition implies that ${A_i}_z\subseteq A_{i+1}$, that is, following again the terminology of
\cite{burstall_rawnsley_1990}, the flag of holomorphic vector bundles \eqref{super} is \emph{super-horizontal}.
\subsection{Normalization of extended solutions}
The following theorem, which is a generalization of Theorem 4.5 in \cite{burstall_guest_1997}, is fundamental to the classification of extended solutions.
\begin{thm}\label{nor}\cite{correia_pacheco_3}
\emph{ Let $\Phi:M\setminus D\to U_\xi(\mathrm{G})$ be an extended solution. Take $\xi'\in \mathfrak{I}'(\mathrm{G})$ such that $\xi\preceq {\xi'}$ and $\g_0^\xi=\g_0^{\xi'}$. Then there exists some constant loop $\gamma\in \Omega_{\mathrm{alg}}\mathrm{G}$ such that $\gamma\Phi:M\setminus D\to U_{\xi'}(\mathrm{G})$.}
\end{thm}
A similar statement holds for extended solutions associated to harmonic maps into symmetric spaces:
\begin{thm}\label{nor2} \emph{ Let $\Phi:M\setminus D\to U^{\mathcal{I}}_\xi(\mathrm{G})$ be an extended solution. Take $\xi'\in \mathfrak{I}'(\mathrm{G})$ such that $\xi\preceq_\mathcal{I} {\xi'}$. Then there exists some constant loop $\gamma\in \Omega^{\mathcal{I}}_{\mathrm{alg}}\mathrm{G}$ such that
$\gamma\Phi:M\setminus D\to U^{\mathcal{I}}_{\xi'}(\mathrm{G})$.}
\end{thm}
\begin{proof}
We can write $\Phi H_+^n=\Psi \gamma_\xi H_+^n$, where $\Psi:M\setminus D\to \Lambda_{\mathrm{alg}}^+G^\C$ contains only even powers of $\lambda$, and consequently $\Psi^{-1}\Psi_z=\sum_{i\geq 0}X'_i\lambda^i$ contains only even powers of $\lambda$. The extended solution equation \eqref{im} gives $\mathrm{Im}\,X'_{2j} \subset \mathfrak{p}^{\xi}_{2j+1}$ for all $j\geq 0$. Set $\hat{\xi}=\xi-\xi'\in \mathfrak{I}^2(\mathrm{G})$. Clearly $\xi\preceq \hat{\xi}$, hence $\mathfrak{p}^\xi_{2j+1}\subseteq \mathfrak{p}^{\hat{\xi}}_{2j+1}$ for all $j\geq 0$. On the other hand, since $\alpha(\hat{\xi})=2\sqrt{-1}\mathbb{Z}$ for any positive root $\alpha$, we have $\mathfrak{g}^{\hat{\xi}}_{2j+1}=0$ and, consequently, $\mathfrak{p}^{\hat{\xi}}_{2j+1}=\mathfrak{p}^{\hat{\xi}}_{2j}$. Hence,
$$\mathrm{Im}\,\Psi^{-1}\Psi_z \subseteq \bigoplus_{j\geq 0} \mathfrak{p}^\xi_{2j+1}\lambda^{2j}\subseteq \bigoplus_{j\geq 0} \mathfrak{p}^{\hat{\xi}}_{2j}\lambda^{2j}\subseteq \Lambda^{+}_{\mathrm{alg}}\mathfrak{g}^\C\cap \gamma_{\hat{\xi}}\Lambda^+\mathfrak{g}^\C\gamma_{\hat{\xi}}^{-1}.$$ Taking account \eqref{holm}, we conclude that $\mathcal{U}_{\xi,\hat{\xi}}(\Phi)$ is anti-holomorphic.
On the other hand, since any extended solution is holomorphic and $\Phi$ is an extended solution, Theorem \ref{popo} asserts that
$\mathcal{U}_{\xi,\hat{\xi}}(\Phi)$ is also holomorphic. Being both holomorphic and anti-holomorphic, it must be equal to a constant loop $\gamma^{-1}$. By Proposition \ref{proposition} we have $\gamma^{-1} \in \Omega^{\mathcal{I}}_{\mathrm{alg}} \mathrm{G}$. Write $\Psi \gamma_{\hat{\xi}}=\gamma^{-1}b$, for some map $b:M\to \Lambda^+\mathrm{G}$.
Then
$$\Phi H^n_+=\Psi \gamma_\xi H^n_+= \gamma^{-1}b \gamma_{\hat{\xi}}^{-1}\gamma_\xi H^n_+= \gamma^{-1}b \gamma_{\xi'} H^n_+,$$
which implies that $\gamma \Phi$ takes values in $U^{\mathcal{I}}_{\xi'}(\mathrm{G})$.
\end{proof}
Given $\xi=\sum n_iH_i$ and $\xi'=\sum n'_iH_i$ in $\mathfrak{I}'(\mathrm{G})$, we have $n_i,n'_i\geq 0$ and observe that $\xi\preceq\xi'$ if and only if $n'_i\leq n_i$ for all $i$.
For each $I\subseteq \{1,\ldots,k\}$, define the cone
$$\mathfrak{C}_{I}=\Big\{\sum_{i=1}^k n_i H_i|\, n_i\geq 0, \,\mbox{$n_j=0$ iff $j\notin I$}\Big\}.$$
\begin{defn}
\emph{Let $\xi\in\mathfrak{I}'(\mathrm{G})\cap \mathfrak{C}_{I}$. We say that $\xi$ is a \emph{$I$-canonical element} of $\mathfrak g$ with respect to $\Delta^+$ if it is a maximal element of $(\mathfrak{I}'(\mathrm{G})\cap \mathfrak{C}_{I},\preceq)$, that is, if $\xi\preceq \xi'$ and $\xi'\in \mathfrak{I}'(\mathrm{G})\cap \mathfrak{C}_{I}$ then $\xi=\xi'$. Similarly, we say that $\xi$ is a \emph{symmetric canonical element} of $\mathfrak g$ with respect to $\Delta^+$ if it is a maximal element of $(\mathfrak{I}'(\mathrm{G}),\preceq_{\mathcal {I}})$ }
\end{defn}
When $G$ has trivial center, which is the case considered in \cite{burstall_guest_1997}, the duals $H_1,\ldots,H_k$ belong to the integer lattice. Then, for each $I$ there exists a unique $I$-canonical element, which is given by $\xi_I=\sum_{i\in I}H_i$.
\begin{thm}
\emph{ Let $\Phi:M \to \Omega_{\rm{alg}}\mathrm{G}$ be an extended solution. There exist a constant loop $\gamma\in \Omega_{\rm{alg}}\mathrm{G}$, a subset $I\subseteq \{1,\ldots,k\}$, a $I$-canonical element $\xi'$ and a discrete subset $D\subset M$, such that
$\gamma\Phi(M\setminus D)\subseteq U_{\xi'}(\mathrm{G})$.}
\end{thm}
\begin{proof}
Take $D\subset M$ and $\xi\in \mathfrak{I}'(\mathrm{G})$ in the conditions of Theorem \ref{usd}. Write $\xi=\sum_{i=1}^kn_iH_i$, with $n_i\geq 0$, and set $I=\{i| n_i>0\}$. By Zorn's lemma, there certainly exists a $I$-canonical element $\xi'$ such that $\xi\preceq \xi'$. On the other hand, from \eqref{gis} we see that $\g_0^\xi=\g_0^{\xi'}$. Hence the result follows from Theorem \ref{nor}. \end{proof}
\begin{thm}
\emph{ Let $\Phi:M \to \Omega^{\mathcal{I}}_{\rm{alg}}\mathrm{G}$ be an extended solution with values in $U_{\xi}^{\mathcal{I}}(\mathrm{G})$, for some $\xi\in\mathfrak{I}'(\mathrm{G})$, off a discrete set $D$. There exist a constant loop $\gamma\in \Omega^{\mathcal{I}}_{\rm{alg}}\mathrm{G}$ and a symmetric canonical element $\xi'$ such that
$\gamma\Phi(M\setminus D)\subseteq U^{\mathcal{I}}_{\xi'}(\mathrm{G})$ and $N_\xi=N_{\xi'}$.}
\end{thm}
\begin{proof}
By Zorn's lemma, there certainly exists a symmetric canonical element $\xi'$ such that $\xi\preceq_\mathcal{I} \xi'$. The result follows from Proposition \ref{proposition} and Theorem \ref{nor2}. \end{proof}
\subsection{Frenet frame data for extended solutions into $\Omega_{\mathrm{alg}}\mathrm{U}(n)$}\label{construction}
Given a finite collection $\{s_j\}$ of meromorphic sections of the trivial bundle $\underline{\C}^n=M\times \C^n$, we obtain an holomorphic bundle away from a discrete subset of $M$, and we can fill in holes to extend it to subbundle $E$ over $M$ of $\underline{\C}^n$. In this case, we denote $E=\mathrm{Span}\{s_j\}$. Reciprocally, any holomorphic subbundle $E$ of $\underline{\C}^n$ has a global meromorphic frame $\{s_1,\ldots, s_k\}$, with $k=\mathrm{rank}\, E$, as explained in \cite{svensson_wood_2010}. For $i>0$, the \emph{$(i)$-th osculating bundle} of $E$ is the subbundle $E^{(i)}$ of $\underline{\C}^n$ spanned by the local holomorphic sections of $E$ and their derivatives up to $i$. We also define the \emph{$(-i)$-th osculating bundle} of $E$ as the subbundle $E^{(-i)}$ of $\underline{\C}^n$ spanned by the local holomorphic sections of $E$ whose derivatives up to $i$ are also local sections of $E$.
Let $g_E=\mathrm{rank}\,E^{(1)}-\mathrm{rank}\,E$ and $r_E$ be the remainder of the positive integer division of $\mathrm{rank}\,E$ by $g_E$: $\mathrm{rank}\,E=q_Eg_E+r_E$.
As Guest \cite{guest_2002} has observed, any smooth map $W:M\to \mathrm{Gr}^n$ corresponding to an extended solution $\Phi:M\to \Omega_{r}^k\mathrm{U}(n)$ is \emph{generated} by a certain holomorphic subbundle $X$, a \emph{Frenet frame} of $\Phi$,
of the trivial bundle $ M\times \lambda^{r}H_+\big/\lambda^kH_+$ by setting
\begin{equation}\label{frenetframe}
W=X+\lambda X^{(1)}+\ldots+\lambda^{k-r-1}X^{(k-r-1)}+\lambda^{k}H_+.
\end{equation}
Hence any extended solution $\Phi:M\to \Omega_{\mathrm{alg}}\mathrm{U}(n)$ can be obtained by applying a finite number of algebraic operations on sets of meromorphic functions on $M$, since $X$ can be chosen arbitrarily. In \cite{ferreira_simoes_wood_2010,svensson_wood_2010} the authors established explicit algebraic formulae relating Frenet frames $X$ with different classes of uniton factorizations of harmonic maps. Next we will give a description of the Frenet frames associated to extended solutions with values in a fixed piece $U_\xi(\mathrm{U}(n))$ of the Bruhat decomposition of $\Omega_{\mathrm{alg}}\mathrm{U}(n)$ and we establish a pure algebraic method to obtain all $\mathrm{S}^1$-invariant extended solutions with values in a fixed $\Omega_\xi$.
Choose a local complex coordinate $z$ and a local section $s$ of $E$. Differentiating $\pi^\perp_E(s)=0$, where $\pi^\perp_E$ is the orthogonal projection onto $E^\perp$, we get $\pi^\perp_E(s_z)=-(\pi_E^\perp)_z(s)$. Hence the association $s \mapsto \pi^\perp_E(s_z)$ defines a local vector bundle morphism $\mathcal{A}_E:E\to E^\perp$, which, following \cite{burstall_wood_1986}, we call the \emph{$\partial'$-second fundamental form} of $E$ in $\C^n$, whose kernel and image do not depend on the choice of the local coordinate $z$. It follows from the linearity of the $\partial'$-second fundamental form that:
\begin{lem}\label{E}
\emph{Let $E$ be a holomorphic vector subbundle of $\underline{\C}^n$.
\begin{enumerate}\item[a)] For all $i\geq 1$, $E^{(-i)}=\ker \mathcal{A}_{E^{(-i+1)}} $ is locally spanned by those sections $s$ of $E$ solving the following system of algebraic linear equations: $(\pi_{E^{(-j)}}^\perp)_z(s)=0$ for all $j=0,\ldots, i-1$;
\item[b)] $ g_{E^{(i)}}\leq g_E$ and $\mathrm{rank}\, E^{(i)}\leq \mathrm{rank}\, E+ig_E$ for all $i\geq 1$ (the equalities hold for $i=1$);
\item[c)] $g_{E^{(-i)}}\leq g_E$ and $\mathrm{rank}\, E^{(-i)}\geq \mathrm{rank}\, E-ig_E$ for all $i\geq 1$(the equalities hold for $i=1$);
\item[d)] For each $g\geq g_E$, there exists a super-horizontal flag of holomorphic subbundles
\begin{equation}\label{E_q}E_{-q}\subsetneq E_{-q+1}\subsetneq \ldots \subsetneq E_{-1}\subsetneq E_0=E,\end{equation} such that $\mathrm{rank}\, E_{-i}=\mathrm{rank}\, E -ig$, where the integer $q\leq q_E$ is the quotient of the positive integer division of $\mathrm{rank}\, E$ by $g$: $\mathrm{rank}\, E=qg+r$.
\end{enumerate}}
\end{lem}
\begin{proof}
For example, since $E^{(-i-1)}=\ker \mathcal{A}_{E^{(-i)}}$, we have, for all $i\geq 0$,
\begin{equation*}\label{n1}
g_{E^{(-i)}}=\mathrm{rank}\,\mathrm{im}\mathcal{A}_{E^{(-i)}}=\mathrm{rank}\,\mathrm{coim}\mathcal{A}_{E^{(-i)}}= \mathrm{rank}E^{(-i)}- \mathrm{rank}E^{(-i-1)}.
\end{equation*}
On the other hand, since
the image of $\mathcal{A}_{E^{(-i-1)}}$ is contained in $ E^{(-i)}\ominus E^{(-i-1)}$, for all $i\geq 0$, we also have
$g_{E^{(-i-1)}}\leq \mathrm{rank}E^{(-i)}- \mathrm{rank}E^{(-i-1)}.$
Hence, for all $i\geq 0$,
$g_{E^{(-i)}}\leq g_E.$
To construct a flag \eqref{E_q}, start by taking an arbitrary holomorphic subbundle $E_{-1}\subseteq E^{(-1)}$ with $\mathrm{rank}\, E_{-1}=\mathrm{rank}\, E -g\leq \mathrm{rank}\, E -g_E=\mathrm{rank}\,E^{(-1)}$. Clearly,
\begin{equation}\label{ge}
g_{E_{-1}}=\mathrm{rank}\,E_{-1}^{(1)}-\mathrm{rank}\,E_{-1}\leq \mathrm{rank}\, E -\mathrm{rank}\,E_{-1}= g.
\end{equation}
Hence $$\mathrm{rank}\, E_{-1}^{(-1)}= \mathrm{rank}\,E_{-1}-g_{E_{-1}}\geq \mathrm{rank}\,E-2g,$$
and we see that there exists a holomorphic subbundle $E_{-2}$ of $E_{-1}^{(-1)}$ with $\mathrm{rank}\,E_{-2} =\mathrm{rank}\,E-2g$. Proceeding recursively we find after $q$ steps a super-horizontal flag of holomorphic subbundles \eqref{E_q}.
\end{proof}
The following construction is fundamental for our purposes:
\begin{lem}\label{algc}
\emph{Let $T\subset E$ be two holomorphic subbundles of $\underline{\C}^n$.
Fix a positive integer $g$, with $g\geq \max\{g_T,g_E\}$, and assume that, for some $i,j\geq 0$, we have $T^{(j)}\subset E^{(-i)}$. Given an integer $d$ with $\mathrm{rank}\,T^{(j)}< d<\mathrm{rank}\,E^{(-i)}$, any holomorphic subbundle $F$ satisfying $T^{(j)}\subset F\subset E^{(-i)}$,
$\mathrm{rank}\,F=d$, and $g_F\leq g$, arises as follows:
\begin{enumerate}
\item[a)] Set
$k_0 =\max\{k\,|\, d-kg> \mathrm{rank}\,T^{(j-k)}\}$ and $r_0 =d-k_0g -\mathrm{rank}\,T^{(j-k_0)}$.
Choose $r_0$ linearly independent meromorphic sections $s_1,\ldots ,s_{r_0}$ of
$E^{(-i-k_0)}$ so that the holomorphic vector bundle
\begin{equation}\label{fk0} F_{-k_0}=T^{(j-k_0)}+\mathrm{Span}\{s_1,\ldots ,s_{r_0}\}\end{equation} has rank $d-k_0g$. Independently of the choice of these meromorphic sections, we have
$g_{F_{-k_0}}\leq g$.
\item[b)] Choose $r_1=d-(k_0-1)g-\mathrm{rank}F_{-k_0}^{(1)}$ meromorphic sections $s_{r_0+1},\ldots ,s_{r_0+r_1}$ of $E^{(-i-k_0+1)}$ so that
the holomorphic vector subbundle
\begin{equation*}\label{fk01}
F_{-k_0+1}=F_{-k_0}^{(1)}+\mathrm{Span}\{s_{r_0+1},\ldots ,s_{r_0+r_1}\}
\end{equation*} has rank $d-(k_0-1)g$. We have $g_{F_{-k_0+1}}\leq g$.
\item[c)] Repeat this procedure $k_0$ times to find a super-horizontal flag of holomorphic subbundles $F_{-k_0}\subsetneq \ldots\subsetneq F_{-1}\subsetneq F_0=F$, with
\begin{equation}\label{k+l}
F_{-k_0+l}=F_{-k_0+l-1}^{(1)}+\mathrm{Span}\{s_{r_0+\ldots+r_{l-1}+1},\ldots ,s_{r_0+\ldots+r_{l-1}+r_l}\},\end{equation}
$r_l=d-(k_0-l)g-\mathrm{rank}F_{-k_0+l-1}^{(1)}$ and $\mathrm{rank}\,F_{-k_0+l}=d-(k_0-l)g.$
\end{enumerate}}
\end{lem}
\begin{proof}
Since $d< \mathrm{rank}\,E^{(-i)}$ and $g_{E^{(-i)}}\leq g_E\leq g$, by Lemma \ref{E}
inequalities $$d-kg< \mathrm{rank}\,E^{(-i)}-kg_{E^{(-i)}}\leq \mathrm{rank}\,E^{(-i-k)}$$ hold for each $k\geq 0$. Hence
we can always take $r_0\geq 0$ linearly independent meromorphic sections of $E^{(-i-k_0)}$ so that ${F_{-k_0}}$ defined by \eqref{fk0} has rank $d-k_0g$. We have to check now that
$g_{F_{-k_0}}\leq g$. By definition of $k_0$ we have
$d-(k_0+1)g\leq \mathrm{rank}\,T^{(j-k_0-1)}.$
Then,
\begin{align*}\label{desg}g_{F_{-k_0}}&\leq g_{T^{(j-k_0)}}+r_0 = g_{T^{(j-k_0)}} +d-k_0g-\mathrm{rank}T^{(j-k_0)} \\ &\leq g_{T^{(j-k_0)}}+g-(\mathrm{rank}T^{(j-k_0)}-\mathrm{rank}T^{(j-k_0-1)})= g_{T^{(j-k_0)}}+g-g_{T^{(j-k_0)}}=g.\end{align*}
Since $F_{-k_0}\subseteq \ker \mathcal{A}_{F_{-k_0+1}}$, then $g_{F_{-k_0+1}}\leq \mathrm{rank}\,{F_{-k_0+1}}-\mathrm{rank}\,{F_{-k_0}}=g$.
On the other hand, it is clear that $r_1\geq 0$. Hence the construction of item b) is possible and we can proceed recursively until find a super-horizontal flag of holomorphic subbundles $F_{-k_0}\subsetneq \ldots\subsetneq F_{-1}\subsetneq F_0=F$, with $F_{-k_0+l}$ given by \eqref{k+l}, where $F$ is certainly in the required conditions.
Reciprocally, any $F$ as required certainly arises in this way. In fact, by Lemma \ref{E} there always exists a super-horizontal flag of holomorphic subbundles $F_{-q}\subsetneq\ldots\subsetneq F_{-k_0}\subsetneq \ldots\subsetneq F_{-1}\subsetneq F_0=F$, with $k_0 =\max\{k\,|\, d-kg> \mathrm{rank}\,T^{(j-k)}\}$. We can choose such sequence so that $T^{(j-k_0)}\subsetneq F_{-k_0}$.
\end{proof}
Now we are in conditions to establish an algorithm to obtain all $\mathrm{S}^1$-invariant extended solutions with values in a given $\Omega_\xi$.
\begin{thm}
\emph{Fix $\xi\in \mathfrak{I}(\mathrm{U}(n))$ and consider the corresponding flag \eqref{flag1}. Set $d_i=\dim A^\xi_i$ and $h_i=d_{i+1}-d_{i}$. Any super-horizontal flag of holomorphic vector subbundles
\begin{equation}\label{flag2}
\{0\}=A_{-r-1}\subsetneq A_{-r} \subseteq \ldots \subseteq A_{k-1} \subsetneq A_k=\underline{\C}^n
\end{equation}
with $\mathrm{rank}\, A_i=d_i$ arises as follows:
\begin{enumerate}
\item[a)] Set $l=\min\{h_i\,|\, i=-r-1,\ldots,k-1\}$ and $m=\max\{i\,|\, l=h_i\}$. Apply Lemma \ref{algc} (with $T=\{0\}$, $E= \underline{\C}^n$, $d= d_m$ and $g=l$) to find $A_m$.
\item[b)] Set
$l_1=\min\{h_i\,|\, -r-1 \leq i <m \}$, $m_1=\max\{i\,|\, l_1=h_i, -r-1 \leq i <m\}$. Apply Lemma \ref{algc} (with $T=\{0\}$, $E= A_m$, $d= d_{m_1}$ and $g=l_1$) to find $A_{m_1}\subseteq {A}_m^{(m_1-m)}.$
\item[c)] Set $l_{\hat{1}}=\min\{h_i\,|\, m<i\leq k-1 \}$ and $m_{\hat{1}}=\max\{i\,|\, l_{\hat{1}}=h_i, m<i\leq k-1\}$, and apply Lemma \ref{algc}
(with $T= A_m$, $E= \underline{\C}^n$, $d= d_{m_{\hat 1}}$ and $g=l_{\hat {1}}$)
to find $A_{m_{\hat{1}}}\supseteq A_m^{(m_{\hat{1}}-m)}.$
\item[d)] Proceed recursively until obtain a flag of the form \eqref{flag2}.
\end{enumerate}}
\end{thm}
\begin{rem}
In \cite{burstall_guest_1997}, the authors introduce a method to obtain super-horizontal flags of holomorphic subspaces associated to a given element $\xi\in\mathfrak{I}'(G)$. However, their method involves integration of meromorphic functions.
\end{rem}
Finally, take a super-horizontal flag of holomorphic vector subbundles \eqref{flag2} and the corresponding $\mathrm{S}^1$-invariant extended solution $W_A$.
Take a meromorphic frame $s_1,\ldots,s_{d_{k-1}}$ of $A_{k-1}$ such that, for each $i\in \{-r,\ldots,k-1\}$, $s_1,\ldots,s_{d_i}$ is a meromorphic frame of $A_i$ and $s_1,\ldots,s_{d_i},s_{d_i+1},\ldots,s_{d_i+g_i}$ is a meromorphic frame of $A_i^{(1)}$. The extended solution
$W$, with values in $U_\xi(\mathrm{U}(n))$ and $u_\xi(W)=W_A$, have Frenet frames of the form
\begin{align}\label{frenetframe}
\nonumber X&= \mathrm{Span}\{s_1\lambda^{-r}+ w_1\lambda^{-r+1},\ldots, s_{d_{-r}}\lambda^{-r}+ w_{d_{-r}}\lambda^{-r+1}\}\\&+\sum_{i=-r}^{k-2}\mathrm{Span}\{s_{d_i+g_i+1}\lambda^{i+1}+w_{d_i+g_i+1}\lambda^{i+2},\ldots,s_{d_{i+1}}\lambda^{i+1}+w_{d_{i+1}}\lambda^{i+2} \};
\end{align}
where, for each $j\in\{1,\ldots,d_{k-1}\}$, $w_j$ is a meromorphic section of
$M\times H_+^n/\lambda^{r+k}H_+^n$. However, in the general case, these meromorphic sections $w_j$ can not be chosen arbitrarily. For example, if $s_1$ is a constant section, $w'_1\lambda^{-r+2}$ becomes a section of $W$. So we have to impose that $p_{0}(w'_1)$, whith $p_0$ the projection defined by \eqref{pi}, has no orthogonal component onto $A^\perp_{-r+2}$. In sections \ref{3s} and \ref{45} we shall discuss in detail some examples.
\section{Extended solutions in $\Omega \mathrm{SU}(n)$}
\subsection{Grassmannian model for $\Omega \mathrm{SU}(n)$}
Consider the exterior product $\wedge$ of vectors in $\C^n$
and extend it to $H^n$ as follows: if $f,g\in H^n$, then $(f\wedge g)(\lambda)=f(\lambda)\wedge g(\lambda)$.
The loop group $\Omega \mathrm{U}(n)$ acts on $\wedge^nH^n$ in the natural way:
$$\gamma(f_1\wedge\ldots\wedge f_n):=\gamma f_1\wedge\ldots\wedge \gamma f_n=\det (\gamma)(f_1\wedge\ldots\wedge f_n).$$
The Grassmannian model of $\Omega \mathrm{SU}(n)$ is given by:
\begin{prop}
\emph{A subspace $W\in \mathrm{Gr}^n$ corresponds to
a loop in $\mathrm{SU}(n)$ if, and only if, it belongs to \bdm
\mathrm{Gr}(\mathrm{SU}(n))=\{W\in \mathrm{Gr}^n\,|\,
\,\wedge^n W=\wedge^n H_+^n\}.\edm}
\end{prop}
\begin{proof}
If $\gamma\in \Omega \mathrm{SU}(n)$, then it is clear that $\wedge^n W=\wedge^n H_+^n$, since $\Omega \mathrm{SU}(n)$ acts trivially on the $\wedge^nH^n$. Conversely, suppose that $\wedge^n W=\wedge^n H_+^n$. For each $\lambda\in \mathrm{S}^1$, consider the isomorphism given by
the evaluation map
at $\lambda$, ${\rm{ev}}_\lambda:W\ominus \lambda W \to \mathbb{C}^n$.
Set $\gamma(\lambda)={\rm{ev}}_\lambda\circ {\rm{ev}}^{-1}_1$, which is a loop of $\Omega \mathrm{U}(n)$ and, by Remark \ref{rem}, verifies $W=\gamma H_+^n$.
By hypothesis
$\wedge^n (W\ominus \lambda W)\subset \wedge^n H_+^n.$
Hence, ${\rm{ev}}_1^{-1}(e_1)\wedge \ldots \wedge {\rm{ev}}_1^{-1}(e_n)\in\wedge^n H_+^n$. Since
\begin{align*}
\det(\gamma)(e_1\wedge\ldots\wedge e_n)&= \gamma(e_1)\wedge\gamma(e_2)\wedge\ldots\wedge \gamma(e_n)={\rm{ev}}_\lambda\circ {\rm{ev}}_1^{-1}(e_1)\wedge \ldots \wedge {\rm{ev}}_\lambda\circ {\rm{ev}}_1^{-1}(e_n)\\&={\rm{ev}}_\lambda( {\rm{ev}}_1^{-1}(e_1)\wedge \ldots \wedge {\rm{ev}}_1^{-1}(e_n) ) ,
\end{align*}
it follows that $\det(\gamma)$ is in $H_+^1$.
Now, since $\wedge^n\gamma H_+^n=\wedge^n H_+^n$, we also have $\wedge^n H_+^n=\wedge^n \gamma^{-1} H_+^n$. Hence, by the same argument as above, $\det(\gamma)^{-1}$ is in $H_+^1$.
On the other hand, the fact that $\gamma$ takes values in $\mathrm{U}(n)$ implies that $\det(\gamma)^{-1}=\overline{\det(\gamma)}$, which means that $\det(\gamma)^{-1}$ is also in $H_-^1$.
This is possible if and only if $\det(\gamma)$ is constant. Since $\gamma(1)=e$, we must have $\det(\gamma)=1$.
\end{proof}
\begin{prop}
If $\xi\in \mathfrak{I}(\mathrm{SU}(n))\subset \mathfrak{I}(\mathrm{U}(n))$, then $U_\xi(\mathrm{SU}(n))=U_\xi(\mathrm{U}(n))$.
\end{prop}
\begin{proof}
Let $\gamma\in U_\xi(\mathrm{U}(n))$. Then $\gamma H^n_+=\Psi \gamma_\xi H^n_+$ for some $\Psi\in \Lambda^+_{\mathrm{alg}}\mathrm{U}(n)$. Hence
\begin{equation}\label{wedge}
\wedge^n\gamma H^n_+ = \wedge^n\Psi\gamma_\xi H^n_+= \wedge ^n\Psi W_\xi=\det (\Psi)\wedge^n W_\xi=\det (\Psi)\wedge^n H^n_+.
\end{equation}
Since $\Psi\in \Lambda^+_{\mathrm{alg}}\mathrm{U}(n)$, $\det (\Psi)$ is polynomial in $\lambda$, hence $\wedge^n\gamma H^n_+\subseteq \wedge^n H^n_+.$ Conversely, from \eqref{wedge} we see that
\begin{equation*}\label{wedge1}
\wedge^n H^n_+=\det (\Psi^{-1})\wedge^n\gamma H^n_+;
\end{equation*}
and since $\det (\Psi^{-1})$ is also polynomial in $\lambda$, we conclude that $\wedge^n H^n_+ \subseteq \wedge^n\gamma H^n_+.$
\end{proof}
In particular, if $\xi\in \mathfrak{I}(\mathrm{SU}(n))$, all the extended solutions $W:M\setminus D\to U_\xi(\mathrm{SU}(n))$ arise from a Frenet frame of the form \eqref{frenetframe} without any further restriction on the choice of the meromorphic data.
\subsection{Canonical elements of $\mathrm{SU}(n)$}
Let $E_i$ be the $(n\times n)$-matrix whose $(i,i)$ entry is $\sqrt{-1}$ and whose other entries are all $0$.
The algebra of diagonal matrices
$$\mathfrak{t}^\C=\big\{\sum a_iE_i:\, a_i\in \C, \, \sum a_i=0\big\}$$
is a
Cartan subalgebra of $\mathfrak{su}(n)^\C=\mathfrak{sl}(n,\C)$.
Let $L_i$ in the dual of $\mathfrak{t}^\C$ be defined by $L_i(\sum a_jE_j)=\sqrt{-1}a_i$.
The corresponding set of roots $\Delta\in \sqrt{-1} \mathfrak{t}^*$ is given by $\Delta=\{L_i-L_j:\, i,j=1,\ldots,n\}$ and $\Delta^+=\{L_i-L_j:\,i<j\}$ is a set of positive roots. The positive simple roots are then the roots of the form $\alpha_i=L_i-L_{i+1}$, with $i=1,\ldots,n-1$, and the dual basis of $\mathfrak{t}$ is formed by the matrices
\begin{align*}
H_i=\frac{n-i}{n}E_1+\ldots+ \frac{n-i}{n}E_i- \frac{i}{n}E_{i+1}-\ldots-\frac{i}{n}E_{n}.
\end{align*}
The Lie group $\mathrm{SU}(n)$ is precisely the simply connected Lie group with Lie algebra $\mathfrak{su}(n)$ and its center is $\mathbb{Z}_n$. The integer lattice $\mathfrak{I}(\mathrm{SU}(n)/\mathbb{Z}_n)$ is simply $\{\sum n_iH_i:\, n_i\in \mathbb{Z}\}$ and its $I$-canonical elements with respect to $\Delta^+$ are the sums $\sum_{i\in I}H_i$, with $I\subseteq \{1,\ldots,n-1\}$. The $I$-canonical elements of $\mathfrak{\mathrm{SU}}(n)$ are not so easy to identify. We need to find the integral combinations of the elements $H_i$ which are in $\mathfrak{I}'(\mathrm{G})\cap \mathfrak{C}_{I}$ (that is, elements which are simultaneously integral combinations of the elements $H_i$ and of the elements $E_i$) and are maximal with respect to $\preceq$. For example,
when $n$ is odd, it is easy to check that
$\xi=H_1+H_2+\ldots + H_{n-1}$ is the unique $[n-1]$-canonical element of $\mathrm{SU}(n)$ with respect to $\Delta^+$, where $[p]=\{1,\ldots,p\}$. But
when $n>2$ is even there always exist more than one $[n-1]$-canonical element. The following lemma is useful in order to describe all canonical elements of $\mathrm{SU}(n)$:
\begin{lem}\label{chis} \emph{The integer lattice $\mathfrak{I}(\mathrm{SU}(n))$ is invariant with respect to the linear isomorphism $\chi_1:\mathfrak{t}^\C\to \mathfrak{t}^\C$ defined by $\chi_1(H_i)=H_{n-i}$, with $i\in[n-1]$.
When $n=2m+1$ is odd, $\mathfrak{I}(\mathrm{SU}(n))$ is also invariant with respect to the linear isomorphism $\chi_2:\mathfrak{t}^\C\to \mathfrak{t}^\C$ defined by $\chi_2(H_i)=H_{2i}$ and $\chi_2(H_{n-i})=H_{n-2i}$ if $i\in\{1,\ldots,m\}$.}
\end{lem}
\begin{proof}
As we have observed before, an element of $\mathfrak{t}$ is in $\mathfrak{I}(\mathrm{SU}(n))$ if and only if its coefficients in $E_i$ are integers. Hence, taking account that $H_i=E_1+\ldots+E_i-\frac{i}{n}(E_1+\ldots+E_n)$, an integer linear combination $\sum_{i=1}^{n-1}n_iH_i$ is in $\mathfrak{I}(\mathrm{SU}(n))$ if and only if $\sum_{i=1}^{n-1}\frac{in_i}{n}$ is an integer number, and this happens if and only if $\sum_{i=1}^{n-1}\frac{(n-i)n_i}{n}$ is integer. Hence $\mathfrak{I}(\mathrm{SU}(n))$ is invariant with respect to $\chi_1$.
Assume now that $n=2m+1$. In this case,
\begin{equation}\label{nis}
\sum_{i=1}^{n-1}\frac{in_i}{n}=\sum_{i=1}^{m}\frac{in_i}{n}+ \sum_{i=1}^{m}\frac{(n-i)n_{n-i}}{n}=\sum_{i=1}^{m}\frac{in_i}{n}- \sum_{i=1}^{m}\frac{in_{n-i}}{n}+\sum_{i=1}^{m}n_{n-i}.
\end{equation}
On the other hand, if we set
\begin{equation*}
\sum_{i=1}^{n-1}n'_iH_i=\chi_2\big(\sum_{i=1}^{n-1}n_iH_i\big)=\sum_{i=1}^mn_iH_{2i}+\sum_{i=1}^mn_{n-i}H_{n-2i},
\end{equation*}
we get
\begin{equation}\label{nis1}
\sum_{i=1}^{n-1}\frac{in'_i}{n}=\sum_{i=1}^m\frac{2i n_i}{n}+\sum_{i=1}^m\frac{(n-2i)n_{n-i}}{n}=2\sum_{i=1}^m\frac{i n_i}{n}-2\sum_{i=1}^m\frac{in_{n-i}}{n}+\sum_{i=1}^{m}n_{n-i}.
\end{equation}
Comparing \eqref{nis} with \eqref{nis1} we conclude that $\sum_{i=1}^{n-1}\frac{in'_i}{n}\in \mathbb{Z}$ if $\sum_{i=1}^{n-1}\frac{in_i}{n}\in \mathbb{Z}$, that is $\mathfrak{I}(\mathrm{SU}(n))$ is invariant with respect to $\chi_2$.
\end{proof}
For each $i\in \{1,\ldots,n-1\}$, let $m_i$ be the least positive integer which makes $m_iH_i$ and integral combination of the elements $E_i$. Since $H_i=E_1+\ldots+E_i-\frac{i}{n}(E_1+\ldots+E_n)$, $m_i$ is precisely the denominator of the irreducible fraction equivalent to $\frac{i}{n}$. The canonical elements should then be sought among the elements of the finite set formed by the integral combinations $\sum_{i=1}^{n-1}n_iH_i$, with $n_i\in\{0,\ldots,m_i\}$, which are simultaneously integral combinations of the elements $E_i$. For general $n$ and $I\subseteq\{1,\ldots,n-1\}$ it is too hard to list all the $I$-canonical elements.
Next we will describe in detail the situation for the lower dimensional cases. We shall denote by $\pi_i$ the orthogonal projection of $\C^n$ onto the one-dimensional vector subspace of $\C^n$ generated by the vector $e_i$.
\subsection{The case $\mathrm{SU}(2)$}
In this case there is a unique simple root $\alpha_1$ with dual $H_1=\frac12(E_1-E_2)$, which does not belong to the integer lattice $\mathfrak{I}(\mathrm{SU}(2))$. Consequently $\xi=2H_1$ is the unique non-trivial canonical element -- the corresponding homomorphism is $\gamma_\xi(\lambda)=\lambda^{-1}\pi_1+\lambda \pi_1^\perp$. If $W:M\setminus D\to U_\xi(\mathrm{SU}(n))$ is a complex extended solution, then the corresponding $\mathrm{S}^1$-invariant solution is given by $u_\xi(W)=\lambda^{-1} A+A+\lambda H_+^2$, where $A$ is a holomorphic subbundle of $\underline{\C}^2$. It follows from the super-horizontality property that $A$ is a constant bundle. Hence,
we have
$W=L+A+\lambda H_+^2$, where $L$ is a holomorphic line bundle of $A\lambda^{-1}+A^\perp$, with $p_{-1}(L)\neq 0$ off a discrete set of points, where $p_{-1}$ is defined as in \eqref{pi}. That is, any harmonic map of finite uniton number from $M$ into $\mathrm{SU}(2)\simeq \mathrm{S}^3$ arises from a constant direction $A$ and a holomorphic line bundle of $\lambda^{-1} A+A^\perp\simeq \underline{\C}^2$. This agrees with the well known result by Calabi \cite{calabi_1967} that asserts that any locally minimal immersion of a surface in an odd dimensional sphere $S^{2m-1}$ is contained in a hyperplane of $\mathbb{R}^{2m}$. In particular, no harmonic map of finite uniton number from $M$ into $\mathrm{SU}(2)\simeq \mathrm{S}^3$ is full. This means that any such harmonic map takes values in a unit two-dimensional sphere $\mathrm{S}^2\simeq \C \mathrm{P}^1$, that is, it corresponds to an holomorphic line bundle of $\underline{\C}^2$.
\subsection{The case $\mathrm{SU}(3)$}\label{3s}
We have two simple roots, $\alpha_1$ and $\alpha_2$, and three non-trivial canonical elements:
\begin{align*}
\xi_1=H_1+H_2=E_1-E_3;\,\,\,\,\,
\xi_2=3H_1=2E_1-E_2-E_3;\,\,\,\,\, \xi_3=3H_2=E_1+E_2-2E_3.
\end{align*}
The corresponding homomorphisms are given by
\begin{align*}
\gamma_{\xi_1}(\lambda)=\lambda^{-1}\pi_3+\pi_2+\lambda \pi_1;\,\,\,\gamma_{\xi_2}(\lambda)=\lambda^{-1}\left(\pi_2+\pi_3\right)+\lambda^2\pi_1;\,\,\,\gamma_{\xi_3}(\lambda)=\lambda^{-2}\pi_3+\lambda(\pi_2+\pi_1).
\end{align*}
If $W_{\xi_1}:M\setminus D\to U_{\xi_1}(\mathrm{SU}(3))$ is a complex extended solution, then the corresponding $\mathrm{S}^1$-invariant solution is given by
$u_{\xi_1}(W_{\xi_1})=\lambda^{-1} B_3+\left(B_2\oplus B_3\right)+\lambda H^3_+,$
where $B_3$ is a holomorphic line subbundle of the holomorphic vector bundle $B_2\oplus B_3$ of rank $2$.
In order to construct all such extended solutions, and taking account the results of section \ref{construction}, we start with a meromorphic section $s_3$ of $\underline{\C}^3$ and set $B_3=\mathrm{Span}\{s_3\}$.
If $B_3$ is not constant, we define $B_{2}=B_3^{(1)}\ominus B_3$, take an arbitrary holomorphic section $w_3$ of $\underline{\C}^3$ and set $X_{\xi_1}=\mathrm{Span}\{\lambda^{-1}s_3+w_3\}$. If $B_3$ is constant, we take an arbitrary meromorphic section $s_2$. By adding a constant if necessary, $s_2$ and $s_3$ are linearly independent and we set $B_2=\mathrm{Span}\{s_2,s_3\}\ominus B_3$. Take an arbitrary holomorphic section $w_3$ of $\underline{\C}^3$ and set $X_{\xi_1}=\mathrm{Span}\{\lambda^{-1}s_3+w_3,s_2\}$. In both cases, $X_{\xi_1}$ is a Frenet frame for an extended solution
$W_{\xi_1}:M\setminus D\to U_{\xi_1}(\mathrm{SU}(3))$.
If $W_{\xi_2}:M\setminus D\to U_{\xi_2}(\mathrm{SU}(3))$ is a complex extended solution, then the corresponding $\mathrm{S}^1$-invariant solution is given by
$$u_{\xi_2}(W_{\xi_2})=\lambda^{-1}(B_2\oplus B_3)+(B_2\oplus B_3)+\lambda(B_2\oplus B_3)+\lambda^2 H_+^3.$$
By the super-horizontality property, $B_2\oplus B_3$ is constant, and consequently $B_1$, the orthogonal complement of $A_2\oplus A_3$, is also constant. In order to construct all such extended solutions, fix a two-dimensional subspace with basis elements $s_2$ and $s_3$. Take arbitrary meromorphic sections $w_2$ and $w_3$ of $\underline{\C}^3+\lambda \underline{\C}^3$ with $\pi_{B_1}(p_0(w_2))$ and $\pi_{B_1}(p_0(w_3))$ constants, where $p_0$ is the projection defined by \eqref{pi}. Then $X_{\xi_2}=\mathrm{Span}\{s_2\lambda^{-1}+w_2, s_3\lambda^{-1}+w_3 \}$ is a Frenet frame for an extended solution
$W_{\xi_2}:M\setminus D\to U_{\xi_2}(\mathrm{SU}(3))$, and all such extended solutions arise in this way.
We observe that, taking account Lemma 3.17 and Proposition 3.18 of \cite{svensson_wood_2010}, the corresponding harmonic map has \emph{uniton number} one, in the sense that it admits an extended solution with values in $\Omega\mathrm{U}(3)$ of the form $\pi_V+\lambda \pi_V^\perp$, with $V$ a holomorphic subbundle of $\underline{\C}^3$.
The case $W_{\xi_3}$ is similar.
\subsection{The cases $\mathrm{SU}(4)$ and $\mathrm{SU}(5)$}\label{45}
Table \ref{cococo} shows all the non-trivial canonical elements of $\mathrm{SU}(4)$ and $\mathrm{SU}(5)$ up to the symmetries $\chi_1,\chi_2$ of Lemma \ref{chis}.
\begin{table}[!htb]
\begin{tabular}{c|c|c|c|c} $\SU(n)$ & $|I|=n-1$ & $|I|=n-2$ & $|I|=n-3$ & $|I|=n-4$\\ \hline $n=4$ &$H_1+2H_2+H_3$ & $2H_1+H_2$ & $4H_1$ & \\ & $3H_1+H_2+H_3$ & $H_1+H_3$ & $2H_2$ & \\ \hline $n=5$ & $H_1+H_2+H_3+H_4$ &$H_1+H_2+4H_3$ & $H_1+2H_2$ & $5H_1$\\ & & $H_1+3H_2+H_3$ & $3H_1+H_2$ & \\ & & $2H_1+H_2+2H_3$ & $H_1+H_4$ &\\ & & $3H_1+2H_2+H_3$ & \\ & & $5H_1+H_2+H_3$ & \end{tabular}
\vspace{.10in}
\caption{Canonical elements for $\mathrm{SU}(4)$ and $\mathrm{SU}(5)$.}\label{cococo}
\end{table}
We describe how to construct, for
$\xi_1=H_1+2H_2+H_3=2E_1+E_2-E_3-2E_4,$
all the extended solutions $W_{\xi_1}:M\setminus D\to U_{\xi_1}(\mathrm{SU}(4))$.
We have $\gamma_{\xi_1}=\lambda^{-2}\pi_4+\lambda^{-1}\pi_3+\lambda\pi_2+\lambda^2\pi_1$, and, consequently,
$$u_{\xi_1}(W_{\xi_1})=\lambda^{-2} B^1_4+\lambda^{-1}\left(B^1_3\oplus B^1_4\right)+\left(B^1_3\oplus B^1_4\right)+\lambda\left(B^1_2\oplus B^1_3\oplus B^1_4\right)+ \lambda^2H_+^4, $$
where each vector subbundle $B^1_i$ has rank one.
The harmonic map associated to this $\mathrm{S}^1$-invariant extended solution is given by
$$\varphi_1=\pi_{B^1_1\oplus B^1_4}-\pi_{B^1_2\oplus B^1_3}.$$
By super-horizontality, $B^1_3\oplus B^1_4$ is a constant bundle. So, in order to construct all such extended solutions, we start by fixing a two-dimensional vector subspace $V$ of $\C^4$ generated by constant vectors $u$ and $v$. Next take meromorphic sections:
\begin{enumerate}
\item $s_1$ of $\underline{V}$, and set $B^1_4=\mathrm{Span}\{s_1\}$ and $B^1_3=\underline{V}\ominus B^1_4$;
\item $s_3$ of $\underline{V}^\perp$, and set $B^1_2=\mathrm{Span}\{s_3\}$ and $B^1_1=\underline{V}^\perp\ominus B^1_2$;
\item $w_1$ of $M\times H^4_+/\lambda^3 H_+^4$.
\end{enumerate} If $B^1_4$ is not constant, we can write
$s_1''=g_1s_1+g_2s'_1$ for some meromorphic functions $g_1$ and $g_2$ on $M$, with $g'_1g_2-g'_2g_1\neq 0$. In this case,
$X=\mathrm{Span}\{\lambda^{-2}s_1+\lambda^{-1}w_1, \lambda s_3\}$ is a Frenet frame for an extended solution with values in $U_{\xi_1}(\mathrm{SU}(4))$ if and only if
$$\pi_{B^1_1}\circ p_0(w_1''-g_1w_1-g_2w_1')=0.$$
For
$\xi_2=3H_1+H_2+H_3=3E_1-E_3-2E_4,$
we have $\gamma_{\xi_2}=\lambda^{-2}\pi_4+\lambda^{-1}\pi_3+\pi_2+\lambda^3\pi_1$, and, consequently,
\begin{align*}
u_{\xi_2}(W_{\xi_2})=\lambda^{-2} B^2_4&+\lambda^{-1}\left(B^2_3\oplus B^2_4\right)\\&+\left(B^2_2\oplus B^2_3\oplus B^2_4\right)+\lambda\left(B^2_2\oplus B^2_3\oplus B^2_4\right)+ \lambda^2\left(B^2_2\oplus B^2_3\oplus B^2_4\right)+\lambda^3 H_+^4. \end{align*}
The harmonic map associated to this $\mathrm{S}^1$-invariant extended solution is given by
$$\varphi_2=\pi_{B^2_2\oplus B^2_4}-\pi_{B^2_1\oplus B^2_3}.$$
Although $\varphi_1$ and $\varphi_2$ are both of the form $\pi_E-\pi_E^\perp$, with $E$ a rank two vector subbundle of $\underline{\C}^4$, these vector bundles exhibit distinct geometrical behaviours. For example, whereas $E=B^2_2\oplus B^2_4$ has always constant $(2)$-osculating bundle, both $E=B^1_1\oplus B^1_4$ and $E^\perp=B^1_2\oplus B^1_3$ can have
non-constant $(2)$-osculating bundle.
\subsection{Symmetric canonical elements of $\mathrm{SU}(n)$}
All the compact inner symmetric spaces of $\mathrm{SU}(n)$ are complex grassmannians $\mathrm{Gr}(k,n)$.
The embedding $\iota$ of $\mathrm{Gr}(k,n)$ as a connected component of $\sqrt{e}$ is given by $\iota (V) =\pi_V - \pi_V^\perp$, where $V\in \mathrm{Gr}(k,n)$ is $k$-dimensional subspace of $\C^n$. There exists no non-trivial symmetric canonical element for $\mathrm{SU}(2)$. Table \ref{xixixi} presents all non-trivial symmetric canonical elements for $\mathrm{SU}(n)$, with $n=3,4,5$, up to the symmetries $\chi_1,\chi_2$.
As before, for each $i\in \{1,\ldots,n-1\}$, let $m_i$ be the least positive integer which makes $m_iH_i$ and integral combination of the elements $E_i$. The symmetric canonical elements should then be sought among the elements of the finite set formed by the integral combinations $\sum_{i=1}^{n-1}n_iH_i$, with $n_i\in\{0,\ldots,2m_i-1\}$, which are simultaneously integral combinations of the elements $E_i$.
We use the usual hermitian inner product on $\C^n$ to identify $\mathrm{Gr}(k,n)$ with $\mathrm{Gr}(n-k,n)$. It is easy to check that, for $\xi\in \mathfrak{I}(\mathrm{SU}(n))$, $N_{\xi}=N_{\chi_1(\xi)}$.
However, in general, the symmetric space $N_{\chi_2(\xi)}$ does not coincide with $N_\xi$. For example, in $\mathrm{SU}(5)$ the two following situations can occur: for $\xi=5H_1$, we have $\chi_2(\xi)=5H_2$, $N_\xi= \mathrm{Gr}(1,5)$ and $N_{\chi_2(\xi)}= \mathrm{Gr}(2,5)$; on the other hand, for $\eta=3H_1+H_2+5H_3$, we have $N_{\eta}=N_{\chi_2(\eta)}=\mathrm{Gr}(2,5)$.
\begin{table}[!htb]
\begin{tabular}{ c| c| c| c| c }
$ \mathrm{Gr}(k,n)$ & $|I|=n-1$ & $|I|=n-2$ & $|I|=n-3$ & $|I|=n-4$\\ \hline
$k=1,n=3$ & $H_1+H_2$ & $3H_1$ & &\\ & $4H_1+H_2$ & & & \\ \hline
$k=2,n=4$ & $3H_1+H_2+H_3$ & $2H_1+H_2$ & & \\ & &$ H_1+H_3$ & & \\
\hline $k=1, n=5$ & $4H_1+2H_2+H_3+H_4$&$ H_1+H_2+4H_3$ & $H_1+2H_2$ & $5H_1$\\ & && $H_1+7H_2$ & \\ && & $3H_1+H_2$&\\ & && $H_1+6H_4$ \\
\hline $k=2, n=5$ & $H_1+H_2+H_3+H_4$ & $H_1+H_2+9H_3$ & $4H_1+3H_2$ & \\ & $2H_1+3H_2+H_3+H_4$ &$H_1+3H_2+H_3$ & $8H_1+H_2$& \\ &$H_1+H_2+H_3+6H_4$ & $H_1+8H_2+H_3$ & $ H_1+H_4$ &\\ & & $2H_1+H_2+2H_3$& \\ & & $3H_1+H_2+5H_3$ & \\ & & $3H_1+2H_2+H_3$ & \\ & & $5H_1+H_2+H_3$ &
\end{tabular}
\vspace{.10in}
\caption{Symmetric canonical elements for $\mathrm{SU}(n)$, with $n\leq 5$.} \label{xixixi}
\end{table}
We describe how to construct, for $n=4$, $k=2$ and
$\xi_1=2H_1+H_2=2E_1-E_3-E_4,$
all the extended solutions $W_{\xi_1}:M\setminus D\to U^{\mathcal{I}}_{\xi_1}(\mathrm{SU}(4))$.
We have $\gamma_{\xi_1}=\lambda^{-1}(\pi_3+\pi_4)+\pi_2+\lambda^2\pi_1$ and, consequently,
$$u_{\xi_1}(W_{\xi_1})=\lambda^{-2} (B^1_4\oplus B^1_3)+(B^1_4\oplus B^1_3\oplus B^1_2)+\lambda (B^1_4\oplus B^1_3\oplus B^1_2)+ \lambda^2H_+^4, $$
where each vector subbundle $B^1_i$ has rank one.
The harmonic map associated to this $\mathrm{S}^1$-invariant extended solution is given by
$\varphi_1=\pi_{B^1_1\oplus B^1_2}-\pi_{B^1_3\oplus B^1_4}.$
So, take meromorphic sections $s_1,s_2,w_1,w_2$ of $\underline{\C}^4$ and set $B_3\oplus B_4=\mathrm{Span}\{s_1,s_2\}$. Assuming that this vector bundle is not constant, $X_{\xi_1}=\mathrm{Span}\{\lambda^{-1}s_1+\lambda w_1,\lambda^{-1}s_2+\lambda w_2\}$ will be a Frenet frame for an extended solution with values in $U^{\mathcal{I}}_{\xi_1}(\mathrm{SU}(4))$. Moreover, all such extended solutions, with $B_3\oplus B_4$ not constant, arise in this way.
Consider also the case $n=4$, $k=2$ and $\xi_2=H_1+H_3=E_1-E_4$. We have $\gamma_{\xi_2}=\lambda^{-1}\pi_4+(\pi_3\oplus\pi_2)+\lambda\pi_1$ and, consequently,
$u_{\xi_2}(W_{\xi_2})=\lambda^{-1} B^2_4+B^2_4\oplus B^2_3\oplus B^2_2+\lambda H_+^4. $
The harmonic map associated to this $\mathrm{S}^1$-invariant extended solution is given by
$\varphi_2=\pi_{B^2_2\oplus B^2_3}-\pi_{B^2_1\oplus B^2_4}.$ Observe that, in this case, we only have $S^1$-invariant extended solutions since $U^{\mathcal{I}}_{\xi_2}(\mathrm{SU}(4))=\Omega_{\xi_2}$.
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"timestamp": "2013-04-17T02:00:31",
"yymm": "1304",
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|
\section{Introduction}
In Classical Mechanics, one of the most venerable equations on a
(connected) Riemannian manifold $(M_0,g_0)$ is:
\[
\hspace*{3.5cm}\frac{D\dot\gamma}{dt}(t)\ =\ - \nabla^{M_0} V (\gamma(t),t)\hspace*{3.5cm}
(E_0)
\]
where $D/dt$ denotes the covariant derivative along $\gamma$
induced by the Levi--Civita connection of $g_0$ and $\dot\gamma$
represents the velocity field along $\gamma$, while $V:M_0\times\R
\rightarrow \R$ is a smooth time--dependent potential. In fact,
when $(M_0,g_0)$ is $\R^3$, this is just Newton's second law for
forces that come from an external time-dependent potential. A
basic property that may have its solutions is {\em completeness}
i.e. the extendability of their domain to all $\R$. At the
beginning of the seventies, some authors studied systematically
this property (see, e.g., \cite{Eb,Go,WM} or also \cite[Theorem
3.7.15]{AM}) but, essentially, they focus only in the autonomous
case, that is, when $V(x,t)\equiv V(x)$ ($V$ is independent of
time).
Very recently, the authors have considered the completeness of the
trajectories not only for the general equation (E$_0$) but also
for more general forces (see \cite{CRS2012}). Concretely, $-
\nabla^{M_0} V$ was generalized to an arbitrary time-dependent
vector field $X$ and forces linearly dependent with the velocity
by means of an operator $F$, were also allowed. Nevertheless, it
is specially interesting to understand and analyze accurately the
differences between the autonomous and the non-autonomous case for
a potential. Moreover, as pointed out in \cite{CFS} (see also
\cite{CaSa}), the completeness for (E$_0$) is equivalent to the
completeness for the geodesics of a class of relativistic
spacetimes that generalizes the classical plane and pp--waves. So,
the aim of the present paper is, first, to analyze further the
completeness in the non-autonomous case $X=- \nabla^{M_0} V$ (even
admitting the linear dependence of the force with the operator
$F$, see equation (E) below) and, then, to analyze the
applications to generalized plane waves.
This paper is organized as follows. In Section 2, we recall the
framework for the completeness of Riemannian trajectories
(Subsection 2.1), and give a new theorem on completeness
(Subsection 2.2). The proofs of two results are provided. The
first one is a technical comparison lemma that is commonly taken
into account in the results on completeness (Lemma
\ref{comparison}). The second one is a theorem on completeness
(Theorem \ref{G01}), obtained by developing further the techniques
in \cite{CRS2012}. In Section 3 we introduce plane wave type
spacetimes (Subsection 3.1) and explain the relation between the
problem of completeness of trajectories and the geodesic
completeness of generalized plane waves (Subsection 3.2).
Moreover, we give further results on geodesic completeness
(Corollaries \ref{complete2}, \ref{complete3}) as a consequence of
the previous result of completeness of trajectories.
\section{Completeness of Riemannian trajectories}
\subsection{Framework}
Let $(\mo,g_0)$ be a (connected) smooth $n$--dimensional
Riemannian manifold and $V:\mo\times\R \rightarrow \R$ a given
smooth function. Taking $p\in \mo$ and $v\in T_p\mo$, there exists
a unique inextensible smooth curve $\gamma : I \to \mo$, $0\in I$,
solution of $(E_0)$ which satisfies the initial conditions
\begin{equation}\label{initial}
\gamma(0) = p,\quad \dot\gamma(0) = v.
\end{equation}
An inextensible solution of $(E_0)$ is {\sl complete} if it is
defined on the whole real line. Note that equation $(E_0)$ in the
trivial case $V\equiv 0$ is the equation of the geodesics in
$(\mo,g_0)$. Let us recall that a Riemannian manifold $(\mo,g_0)$
is {\sl geodesically complete} if any of its inextensible
geodesics is defined on $\mathbb{R}$ or, equivalently, the metric
distance induced by $g_0$ is complete.
In \cite[Theorem 2.1]{Go} Gordon proved the completeness of the
trajectories of $(E_0)$ if the potential $V$ is time--independent,
bounded from below and satisfying either $(\mo,g_0)$ is complete
or $V$ is proper (i.e., $V^{-1}(K)$ is compact in $\mo$ for any
compact $K\subset \R$). Other results in the autonomous case were
given in \cite{Eb,WM} and \cite[Theorem 3.7.15]{AM}.
Following \cite{CRS2012}, we generalize such results to the
non--autonomous case by including also the action of a (1,1)
tensor field $F$ along the natural projection $\pi : \mo \times \R
\longrightarrow \mo$, i.e., we consider the second order
differential equation
\[
\hspace*{2.9cm}\frac{D\dot\gamma}{dt}(t)\ =\ F_{(\gamma(t),t)}\
\dot\gamma(t) - \nabla^{M_0} V (\gamma(t),t).\hspace*{2cm}(E)
\]
Let us remark that the existence and uniqueness result of
inextensible solutions of $(E)$, under the same initial conditions
\eqref{initial}, remains now true, and, obviously, one has the notion of
complete inextensible trajectory of $(E)$.
Now, let us introduce some terminology in order to express natural
conditions on $F$ and $V$. Notice that, in general, $F$ is
neither self-adjoint nor skew-adjoint with respect to $g_0$, and
denote by $S$ the self--adjoint part of $F$.
For each $t \in \R$, put
\[
\| S(t) \|\ : = \
\max\big\{\big|S_{\sup}(t)\big|,\, \big|S_{\inf}(t)\big|\big\}
\]
where
\[
S_{\sup}(t): = \sup_{\underset{\|v\|=1}{v\in T\mo}}
g\left(v,S_{(p,t)} v\right) \quad \text{and} \quad S_{\inf}(t):=
\inf_{\underset{\|v\|=1}{v\in T\mo}} g\left(v,S_{(p,t)} v\right).
\]
We say that $S$ is {\em bounded} (resp. {\em upper bounded}, {\em
lower bounded}) {\em along finite times} when, for each $T>0$,
there exists a constant $N_T$ such that
\begin{equation}\label{bf}
\|S(t)\| \le N_T \; \hbox{(resp. $S_{\sup}(t) \le N_T$, $-S_{\inf}(t) \le N_T$)}\; \hbox{for all $t\in [-T,T]$.}
\end{equation}
Moreover, the potential $V$ is {\em bounded from below along
finite times} if there exists a continuous function $\beta_0:
\R\rightarrow \R$ such that
\begin{equation}\label{bf1}
V(p,t)\ \ge\ \beta_0(t)\quad \hbox{for all}\quad (p,t)\in \mo
\times \R.
\end{equation}
In order to investigate the completeness of the inextensible
solutions of equation $(E)$, let us recall that an integral curve
$\rho$ of a vector field on a manifold, defined on some bounded
interval $[a,b)$, $b<+\infty$, can be extended to $b$ (as an
integral curve) if and only if there exists a sequence
$\{t_n\}_n$, $t_n \to b^-$, such that $\{\rho(t_n)\}_n$ converges
\cite[Lemma 1.56]{ON}. The following technical result follows
directly from this fact and \cite[Lemma 3.1]{CRS2012}.
\begin{lemma}\label{extend}
Let $\gamma: [0,b) \to \mo$ be a solution of equation $(E)$ with
$0<b<+\infty$. The curve $\gamma$ can be extended to $b$ as a
solution of $(E)$ if and only if there exists a sequence
$\{t_n\}_n \subset [0,b)$ such that $t_n \to b^-$ and the sequence
of velocities $\{\dot\gamma(t_n)\}_n$ is convergent in the tangent
bundle $T\mo$.
\end{lemma}
Furthermore, we need also the following result (compare with
\cite[Example 2.2.H]{AM}).
\begin{lemma}[{\bf Comparison Lemma}]\label{comparison}
Let $\varphi :[a,+\infty) \to \R$ be a continuous
monotone increasing function such that
\begin{equation}\label{et1}
\varphi(s) > 0\quad\ \hbox{for all $\ s\ge a$}
\quad\hbox{and}\quad \int_a^{+\infty} \frac{ds}{\varphi(s)} =
+\infty.
\end{equation}
If a $C^1$ function $v_0=v_0(t)$ satisfies the equation
\begin{equation}\label{global}
v_0'(t)\ =\ \varphi(v_0(t))\quad \hbox{with $v_0(0) \ge a$,}
\end{equation}
and it is inextensible, then it is defined for all $t \ge 0$.
\vspace{1mm}
Furthermore, if $v :[0,b) \to \R$ is a continuous function such
that
\begin{equation}\label{stime}
\left\{\begin{array}{ll} \displaystyle a \le v(t) \le v(0) + \int_0^t
\varphi(v(s)) d s &\hbox{for all \;
$t \in [0,b)$,}\\[1mm]
v(0) \le v_0(0),&
\end{array}\right.
\end{equation}
then $v(t) \le v_0(t)$ for all $t \in [0,b)$.
\end{lemma}
\begin{proof} Even though this is a simple exercise, we
prefer to give here a complete argument by completeness. If
$v_0=v_0(t)$ is a $C^1$ inextensible solution of \eqref{global} in
the interval $[0,\bar b)$, then
\begin{equation}\label{et2}
v_0(t) \ge v_0(0) \ge a\quad \hbox{for all $t\in [0,\bar b)$,}
\end{equation}
whence, for all $t\in [0,\bar b)$, $\varphi(v_0(t))(>0)$ is well
defined and $v_0$ becomes strictly monotone increasing. Thus,
dividing both the terms of \eqref{global} by $\varphi(v_0(t))$ and
integrating in $[0,t]$, $0 < t < \bar b$, we have
\[
\int_0^t\frac{v_0'(\tau)}{\varphi(v_0(\tau))}d\tau = t,
\]
hence, $v_0 = v_0(t)$ is the inverse of
\begin{equation}\label{et}
t(v_0)\ =\ \int_{v_0(0)}^{v_0} \frac{ds}{\varphi(s)},
\end{equation}
with the maximum $\bar b$ equal to
$\displaystyle\lim_{v_0\rightarrow +\infty}t(v_0)$ in \eqref{et}.
From \eqref{et1} it follows $\bar b = +\infty$.
\vspace{1mm}
Now, let $v=v(t)$, $t\in [0,b)$, be such to satisfy \eqref{stime}
and define
\[
h(t) \ =\ v_0(0) +\ \int_0^t \varphi(v(s)) ds.
\]
Clearly, $h$ is a $C^1$ function such that
\[
h(0) = v_0(0)\quad\hbox{and}\quad h'(t) = \varphi(v(t))
\quad \hbox{for all $t\in [0,b)$.}
\]
Moreover, from \eqref{stime} it follows
\begin{equation}\label{stima3}
a\ \le\ v(t) \ \le\ h(t)\quad \hbox{for all $t\in [0,b)$,}
\end{equation}
whence the monotonicity of $\varphi$ implies
\begin{equation}\label{stima2}
h'(t)\ \le\ \varphi(h(t))\quad \hbox{for all $t\in [0,b)$.}
\end{equation}
Thus, from \eqref{et1}, \eqref{global} and \eqref{stima2} we have
\[
\frac{h'(t)}{\varphi(h(t))} \ \le\ 1\ =\
\frac{v_0'(t)}{\varphi(v_0(t))} \quad \hbox{for all $t\in [0,b)$,}
\]
whence direct computations give
\begin{equation}\label{et4}
\int_{v_0(0)}^{h(t)}\frac{ds}{\varphi(s)} \ \le\
\int_{v_0(0)}^{v_0(t)}\frac{ds}{\varphi(s)}
\quad \hbox{for all $t\in [0,b)$.}
\end{equation}
Now, assume that
$\bar t \in (0,b)$ exists such that $h(\bar t) > v_0(\bar t)$.
Hence, \eqref{et1} and \eqref{et2} imply
\[
\int_{v_0(0)}^{h(\bar t)}\frac{ds}{\varphi(s)} \ >\
\int_{v_0(0)}^{v_0(\bar t)}\frac{ds}{\varphi(s)}
\]
in contradiction with \eqref{et4}. So, we have
$h(t) \le v_0(t)$ for all $t \in [0,b)$ and the proof follows from \eqref{stima3}.
\end{proof}
\subsection{Our main result on the non--autonomous problem $(E)$}
Now, we are ready to state our main result on the completeness of inextensible trajectories
of the non--autonomous problem $(E)$.
\begin{theorem}\label{G01} Let $(\mo,\g)$ be a complete Riemannian mani\-fold, $F$ a
smooth time--dependent $(1,1)$ tensor field with self--adjoint
component $S$ and $V:\mo\times\R \rightarrow \R$ a smooth
potential. Assume that $\|S(t)\|$ is bounded along finite times,
$V$ is bounded from below along finite times
and there exists a continuous function $\alpha_0: \R\rightarrow \R$
such that
\[
\left|\frac{\partial V}{\partial t}(p,t)\right|\ \le\ \alpha_0(t)
(V(p,t) - \beta_0(t))\quad \hbox{for all \, $(p,t)\in \mo\times \R$}
\]
with $\beta_0$ as in \eqref{bf1}.
Then, each inextensible solution of equation $(E)$ must be complete.
\end{theorem}
The proof of Theorem \ref{G01} is a direct consequence
of the following more general result.
\begin{proposition}\label{G011} Let $(\mo,\g)$ be a complete Riemannian manifold, $F$ a
smooth time--dependent $(1,1)$ tensor field with self--adjoint
component $S$ and $V:\mo\times\R \rightarrow \R$ a smooth
potential bounded from below along finite times with $\beta_0$ as in \eqref{bf1}.
\vspace{1mm}
If $S_{\sup}(t)$ is upper bounded along finite times
and a continuous function $\alpha_0: \R\rightarrow \R$ exists
such that
\[
\frac{\partial V}{\partial t}(p,t)\ \le\ \alpha_0(t)
(V(p,t) - \beta_0(t))\quad \hbox{for all \, $(p,t)\in \mo\times
\R$,}
\]
then each inextensible solution of equation $(E)$ must be forward
complete.
\vspace{1mm}
Conversely, if $S_{\inf}(t)$ is lower bounded along finite times
and a continuous function $\alpha_0: \R\rightarrow \R$ exists such
that
\[
- \frac{\partial V}{\partial t}(p,t)\ \le\ \alpha_0(t)
(V(p,t) - \beta_0(t))\quad \hbox{for all \, $(p,t)\in \mo\times
\R$,}
\]
then each inextensible solution of equation $(E)$ must be backward
complete.
\end{proposition}
\begin{proof}
Let $\gamma$ be a non--constant forward inextensible solution of
equation $(E)$ defined on the interval $[0,b) \subset \R$. Arguing
by contradiction, assume that $\gamma$ is not forward complete,
i.e., $b<+\infty$, so a real positive constant $T > b$ can be
fixed so that \eqref{bf} holds for $S_{\sup}(t)$, furthermore
\begin{equation}\label{stima5}
V(p,t) - B_T \ge 1\quad \hbox{and}\quad
\frac{\partial V}{\partial t}(p,t)\ \le\ A_T
(V(p,t) - B_T)
\end{equation}
for all \, $(p,t)\in \mo\times [-T,T]$, with $A_T \ge \max \alpha_0([-T,T])$
and $B_T \le \min\beta_0([-T,T]) -1$.
Now, for simplicity, denote
\[
u(t) = g(\dot\gamma(t),\dot\gamma(t))\;\; \hbox{and}\;\;
v(t)\ =\ \frac12\ u(t) + V(\gamma(t),t) - B_T,
\quad t\in [0,b).
\]
From \eqref{stima5} it follows
\[
u(t)+1\ \le\ 2 v(t),
\]
hence if $v(t)$ is bounded in $[0,b)$ so is $u(t)$, that is
a constant $k > 0$ exists such that
\begin{equation}\label{inequality3}
u(t)\ \le\ k \quad\hbox{for all $t \in [0,b)$.}
\end{equation}
Note that this inequality is enough for contradicting that $b$ is
finite. In fact, \eqref{inequality3} implies that
$\dot\gamma([0,b))$ is bounded in $T\mo$ and, being $(\mo,\g)$
complete, Lemma \ref{extend} is applicable because of the
completeness of $M_0$. Hence, $\gamma$ can be extended to $b$ in
contradiction with its maximality assumption.
In order to prove that $v(t)$ is bounded in $[0,b)$,
taking any $t\in [0,b)$ by using equation $(E)$
and estimates \eqref{bf} and \eqref{stima5} we have
\[\begin{split}
\frac{d v}{dt}(t)\ &=\
g\big(\frac{D\gamma}{dt}(t),\dot\gamma(t)\big)
+ g\left(\nabla^{\mo}V(\gamma(t),t),\dot\gamma(t)\right) + \frac{\partial V}{\partial t}(\gamma(t),t)\\
&=\ g\big(F_{(\gamma(t),t)}\dot \gamma(t),\dot\gamma(t)\big)\, + \,\frac{\partial V}{\partial
t}(\gamma(t),t)\\[1mm]
&=\ g\big(S_{(\gamma(t),t)}\dot \gamma(t),\dot\gamma(t)\big)\, + \,\frac{\partial V}{\partial
t}(\gamma(t),t)\\[1mm]
&\le\ N_T\, u(t) + A_T \big(V(\gamma(t),t)-B_T\big).
\end{split}
\]
Whence, $A_T^* \in \R$ exists such that
\begin{equation}\label{G05}
\frac{d v}{dt}(t)\ \le\ A_T^*\, v(t) \quad\hbox{for all $t \in
[0,b)$.}
\end{equation}
On the other hand, if we consider the linear equation
\begin{equation}\label{G06}
w'(t)\ =\ A_T^*\ w(t),
\end{equation}
let $v_0=v_0(t)$ be the unique (global) solution of \eqref{G06}
satisfying the initial condition $v_0(0) = v(0)$, with $v(0) \ge 1$ from \eqref{stima5}.
Thus, from \eqref{G05} and Lemma \ref{comparison} with $\varphi(s) = A_T^* s$
and $a=1$, we have that $v(t) \le v_0(t)$ for all $t\in [0,b)$, with $v_0(t)$
bounded in $[0,b]$; whence, $v(t)$ is bounded in $[0,b)$.
\vspace{1mm}
Conversely, let us assume that $\gamma$ is not backward complete
in $(-b,0]$ with $b < +\infty$, then we can consider $T>b$ and
$\tilde \gamma(t) := \gamma(-t)$ in $[0,b)$. From the lower
boundedness of $S_{\inf}(t)$ in $[-T,T]$ and the estimate on
$-\frac{\partial V}{\partial t}$ along finite times, we have
\[
\begin{split}
\frac{dv}{dt}(-t)\ &=\
- g\left(S_{(\gamma(-t),-t)} \dot\gamma(-t),\dot\gamma(-t)\right)
- \frac{\partial V}{\partial t}(\gamma(-t),-t)\\[1mm]
&\le\ N_T\, u(-t) + A_T \big(V(\gamma(-t),-t) - B_T\big),
\end{split}
\]
and we repeat the above argument for $\tilde\gamma(t)$.
\end{proof}
\begin{remark}
Both in Theorem \ref{G01} and in Proposition \ref{G011} the
assumption on the completeness of $(\mo,\g)$ can be replaced by
the condition ``{\sl $V$ is proper}''. In fact, in the above proof
once we have proven that $v(t)$ is bounded in $[0,b)$, the
properness of $V$ implies that $\dot \gamma ([0,b))$ lies in a
compact subset of $T\mo$, so $\gamma$ can be extended to $b$.
\end{remark}
\begin{remark}
As commented in the Introduction, other completeness results on
the inextensible trajectories of equation $(E)$ as well as their
comparison with Theorem \ref{G01} can be found in \cite{CRS2012}.
\end{remark}
\section{Geodesic completeness of GPW}
\subsection{Plane waves and their generalizations}
A {\em parallely propagated wave} spacetime, or a {\em pp--wave} in
brief, is a relativistic spacetime $(\R^4,ds^2)$ where the
Lorentzian metric $ds^2$ has the form
\[
ds^2 \ =\ dx^2+dy^2 + 2dudv + H(x,y,u) du^2,
\]
being $(x,y,u,v)$ the natural coordinates of $\R^4$ and $H : \R^3
\to \R$ a non--zero smooth function. If the expression of $H$ is
quadratic in $x, y$, i.e.,
\begin{equation} \label{ehpw}
H(x,y,u)\ =\ f_1(u) x^2 - f_2(u) y^2 + 2 f(u) xy,
\end{equation}
for some smooth real functions $f_1$, $f_2$ and $f$, then the
spacetime is called {\em plane wave}, and, in particular, an {\sl
(exact plane fronted) gravitational wave} if $f_1 \equiv f_2$ (for example, see \cite{BEE}).
Since the pioneer papers dealing with gravitational waves
\cite{Br,ER}, these spacetimes have been widely studied by many
authors (see \cite{CFS} and references therein or the summary in
\cite{Yu}) not only for their geometric interest but above all for
their physical interpretation. In fact, as explained in
\cite{MTW}, a gravitational wave represents ripples in the shape
of spacetime which propagate across spacetime, as water waves are
small ripples in the shape of the ocean's surface propagating
across the ocean. The source of a gravitational wave is the motion
of massive particles; in order to be detectable, very massive
objects under violent dynamics must be involved (binary stars,
supernovas, gravitational collapses of stars...). With more
generality, pp--waves may also taken into account the propagation
of non--gravitational effects such as electromagnetism.
Here, we focus only on the property of geodesic
completeness. In particular, we add further information to
the study of the geometric properties for the family of
generalized plane waves, already developed in \cite{CFS,
CRS2012, FS_CQG, FS_JHEP}. The key fact is that the geodesic completeness
of a pp--wave reduces to the completeness of the inextensible
trajectories that are solutions of the second order differential
equation (E$_0$) when $(M_0,g_0)$ is $\R^2$. However, this last
restriction is not important and, following \cite{FS_CQG}, the
classical notion of pp--wave can be generalized as follows:
\begin{definition}\label{pfwave}
{\rm A Lorentzian manifold $(\m,g)$ is called \emph{generalized
plane wave}, briefly \emph{GPW}, if there exists a connected
$n$--dimensional Riemannian manifold $(\mo,\g)$ such that $\m =
\mo \times \R^{2}$ and
\begin{equation}\label{wave}
g\ =\ \g + 2dudv+ \h (x,u) du^{2},
\end{equation}
where $x\in \mo$, the variables $(u,v)$ are the natural
coordinates of $\R^{2}$ and the smooth function $\h: \mo\times
\R\rightarrow \R$ is such that $\h\not\equiv 0 $.}
\end{definition}
\subsection{Application to geodesic completeness}
In order to investigate the properties of geodesics in a GPW, it
is enough studying the behavior of the Riemannian trajectories
under a suitable potential $V$. In particular, the problem of
geodesic completeness is fully reduced to a purely Riemannian
problem: the completeness of the inextensible trajectories of
particles moving under the potential $V(x,u) = - \frac{1}2\,
\h(x,u)$ as the following result shows (see \cite[Theorem
3.2]{CFS} for more details).
\begin{theorem}
\label{traj} A GPW is geodesically complete if and only if
$(\mo,\g) $ is a complete Riemannian manifold and the inextensible
trajectories of
\[
\hspace*{45mm}\frac{D\dot{\gamma}}{dt}\ =\
\frac{1}{2}\,\nabla^{\mo}\h(\gamma(t),t) \hspace*{35mm} (E_0^*)
\]
are complete.
\end{theorem}
Now, we can use Theorem \ref{G01} to obtain the completeness of
the inextensible trajectories of equation $(E_0^*)$. Then, the
following result on the geodesic completeness on GPW can be
stated:
\begin{corollary}
\label{complete2} Let $\m=\mo\times\R^{2}$ be a GPW such that
$(\mo,\g)$ is a geodesically complete Riemannian manifold and
$\h:\mo\times \R\rightarrow \R$ is a smooth function. If there
exist two continuous functions $\alpha_0$, $\beta_0: \R\rightarrow
\R$ such that
\[
\h(x,u) \ \le\ \beta_0(u)\quad \hbox{and} \quad
\left|\frac{\partial \h}{\partial u}(x,u)\right|\ \le\ \alpha_0(u)
\left(\beta_0(u) - \h(x,u)\right)
\]
for all $(x,u) \in \mo \times\R$, then $(\m,g)$ is geodesically
complete.
\end{corollary}
We emphasize that other results on autonomous and non-autonomous
potentials can be translated into results of geodesic completeness
of GPW. So, as a consequence of \cite[Corollary 3.6]{CRS2012} we
have:
\begin{corollary}
\label{complete3} A GPW with complete $(\mo,\g)$ is geodesically
complete if $\nabla^M\h$ grows at most linearly in $M$ along
finite times
\end{corollary}
\begin{remark}
The particular case of this corollary for pp--waves (i.e. its
application for $(\mo,\g)=\R^2$) was discussed in \cite{CRS2012},
and it has a clear interpretation: not only classical plane waves
are geodesically complete but also each pp--wave such that its
coefficient $\h$ behaves qualitatively as the one of a plane wave,
are. This can be understood as a result of stability of the
completeness of plane waves in the class of all pp--waves. So,
Corollary \ref{complete3} also ensures stability of completeness
in the class of generalized plane waves.
Even though the physical interpretation of Corollary
\ref{complete2} is not so clear, it is logically independent of
Corollary \ref{complete3} (a discussion as the one below
Proposition 3.7 in \cite{CRS2012} also holds here). This shows
that the application of the techniques are not exhausted and,
under motivated assumptions, further results could be obtained.
\end{remark}
|
{
"timestamp": "2013-04-18T02:02:26",
"yymm": "1304",
"arxiv_id": "1304.4818",
"language": "en",
"url": "https://arxiv.org/abs/1304.4818"
}
|
\section*{Introduction}
The identification and analysis of covarying positions in a protein family gives important insights into that family's evolutionary history and provides information about sites that are important for function and structural stability as it is believed that covariation implies coevolution \cite{Atchley:2000,Tillier:2003p171,Gloor:2005vu,Travers:2007}. Coevolutionary analysis of protein families is important because it potentially provides a direct link between primary sequence, in the form of multiple sequence alignments, and structure/function predictions. Covariation between positions in a protein family is assumed to derive from phylogenetic, structural, functional, interaction, and stochastic signals \cite{Atchley:2000}. Decomposing this signal is difficult because the phylogenetic and stochastic signal can overwhelm the structural and functional signal \cite{Martin:2005}. Furthermore, alignment errors have been shown to produce misleading erroneous signal \cite{Dickson:2010p237}.
One of the most popular methods for quantifying covariation in proteins is Mutual Information ($MI$). There are many coevolution prediction methods which are derived from MI \cite{Gloor:2005vu,Dunn:2008,Little:2009,Buslje:2009ct,Dickson:2010p237}. As well, there are many web-based servers which will calculate Mutual Information from a submitted protein alignment \cite{Kozma:uj,Chakraborty:2012cb,Yip:2008,GouveiaOliveira:2009eb}. Despite its simple formulation, calculation of MI is computationally demanding, largely because it must be calculated for all pairs of positions in the alignment, meaning it scales $n^2$ relative to the length of the alignment. Further, calculating inter-protein coevolution requires concatenated alignments which increases the effective number of pairs of positions.
Herein we describe an algorithm for calculating MI in protein alignments with high efficiency. This algorithm allows for database-wide analysis \cite{Dickson:2010p237} and real-time calculation of covariation during alignment curation \cite{Dickson:2012jx}. This algorithm is included as part of the MIpToolset.
\section*{Algorithm}
\subsection*{Mutual Information}
The calculation and formulation of Mutual Information is described in detail in \cite{Martin:2005}; it is outlined here to provide necessary background to understand the optimizations of the MIpToolset algorithm.
Mutual Information measures the degree of covariation between two random variables (in our case, protein alignment positions $X$ and $Y$) using the Information Theoretic quantity Entropy ($H$).
\begin{equation}\label{MI}
MI_{x,y} = H_x+ H_y - H_{x,y}
\end{equation}
Information Entropy ($H$) can be understood as the measure of uncertainty of the identity of the amino acid at some position $x$. As shown in equation \ref{entropy}, the Entropy ($H$) for position $x$ is calculated using the probability of each of the 20 amino acids appearing at that position. Since the actual probabilities are unknown, the amino acid frequencies in the input alignment are used to approximate these values.
\begin{equation}\label{entropy}
H_x= -\sum_{i=1}^{20} p(x_i)\log_{20} p(x_i)
\end{equation}
The $MI$ between positions $X$ and $Y$ is the sum of the Entropy of each position minus the "joint Entropy" between them. The enumeration of joint entropy is the rate-limiting step of Mutual Information calculations. Joint Entropy is calculated similarly to Entropy, but it involves the calculation of probability of all pairs of amino acids that occur between position $x$ and position $y$ (Equation \ref{jointentropy}).
\begin{equation}\label{jointentropy}
H_{x,y}= -\sum_{i=1}^{20}\sum_{j=1}^{20} p(x_i, y_j)\log_{20} p(x_i, y_j)
\end{equation}
The na\"{\i}ve calculation of joint entropy is inefficient because it involves populating a 20 x 20 matrix for every pair of amino acids found for every pair of positions. This is a 400-entry matrix for $n^2$ positions. This approach, while easy to implement, uses an unnecessary amount of memory as it does not exploit the fact that most positions will be moderately conserved and, thus, most positions will have a value of zero in the joint entropy count matrix.
\subsection*{Storage of sparse matrix in linked list}
\begin{figure}[tbh]
\centerline{\includegraphics{fig1.eps}}
\caption{{\bf The linked list storage of amino acid pair counts.} This figure demonstrates how two new amino acid pairs, LS and LH, are added to the growing linked list data structure which stores amino acid pair counts. First, LS is added to the list by incrementing the existing LS node. Second, the pair LH is added to the list by creating a new node labeled LH and adding it to the list with counter set to 1. }
\label{fig1}
\end{figure}
It is worth noting that calculation of $MI$ involves two types of "pairs": Pairs of $positions$, which represent the homologous `columns' in a protein family multiple sequence alignment (MSA), and pairs of $amino~acids$, which are the corresponding entries from a pair of positions within a single sequence. So a pair of positions, might be position 10 and position 45 within a protein sequence; at this pair of positions, there will be many amino acid pairs corresponding to the identity of the amino acids at positions 10 and 45 in the sequence (ie. DF, LS, DH etc.).
A straightforward way to store the counts between positions $x$ and $y$ is to use a linked list data structure (Figure \ref{fig1}). Each node in the linked list stores two values for the calculation of Joint Entropy, the identity of the amino acid pair, and the respective count. Each node also contains a pointer to the next node in the list, or $null$ if the node is the terminal node.
The program iterates over the protein alignment, enumerating the amino acid pairs, just as it would if it were in the na\"{\i}ve implementation. If an entry in the linked list exists for a given amino acid pair, the node's counter is incremented. If no such entry exists a new node is appended to the end of the list for that amino acid pair. This list can be traversed efficiently as these counts will be used for future calculations. This efficient storage makes it possible to efficiently analyze very long alignments.
\subsection*{Direct access to linked list improves speed}
\begin{figure}[tbh]
\centerline{\includegraphics[width=0.8\textwidth]{fig2.eps}}
\caption{{\bf Direct-access array of pointers to growing linked list.} This figure demonstrates how the direct-access array provides instant access to any part of the linked list without the need to traverse the list. This array is only allocated once and can be reused for each pair of positions.}
\label{fig2}
\end{figure}
The limitation of the linked list storage method, if a linked list is used on its own, is that the list will need to be traversed each time a node is to be updated or created to check whether that pair exists in the data structure. This challenge can be overcome by using an array of pointers to linked list nodes. The disadvantage of the linked list storage solution is that it lacks "direct access" provided by a two-dimensional array in from the na\"{\i}ve implementation. By combining the two, it is possible to achieve a ``best of both worlds'' solution.
A single "direct-access" 20 x 20 array is created, with the nodes in the array corresponding to the 400 possible amino acid pairs (Figure \ref{fig2}). When an amino acid pair is encountered by the main count enumeration loop, the direct-access array is checked. If the entry for that pair is $null$, then a new linked node is appended to the end of the growing linked list for that pair of positions with a count of 1; next, the entry in the direct-access matrix is set as a pointer to the newly created linked list node.
Conversely, if the entry corresponding to the amino acid pair contains a pointer, the program follows the pointer to the corresponding linked list node and increments the counter by 1. After the two positions have been fully enumerated, all entries in the direct-access array are reset to $null$ and it can be reused. Thus, the direct-access array strategy maintains the advantages of a linked list storage solution without the disadvantage of needing to traverse the list every time, at the trade-off cost of only 400 pointers.
\subsection*{Integration in the MIpToolset}
This algorithm has been included as part of the MIpToolset, a collection of C- and Perl-based programs which calculate covariation statistics and inter-residue distances from protein alignments and databases. A full description of sequence collection and alignment is available in (Dickson and Gloor, Methods Mol. Biol. 2013 $submitted$).
In brief, the input to the program is a protein alignment containing more than 150 sequences less than 90\% identical and containing more than 50 ungapped positions. It is recommended that the alignment be manually analyzed by the investigator to ensure the alignment does not contain errors which will lead to false-positive results \cite{Dickson:2010p237,Dickson:2012jx}. For example, the curation tool LoCo \cite{Dickson:2012jx}, based on the alignment viewer Jalview \cite{Waterhouse:2009}, provides a visualization of the likely-misaligned regions of the alignment. The program also optionally accepts a PDB structure corresponding to a sequence in the protein family. This structure is used to generate inter-residue distances which are commonly used to validate coevolution predictions.
The output of the program is a large list of pairs of positions and their corresponding covariation statistics. The MIpToolset presently generates Mutual Information, as well as several more accurate derivations including $MIp$ (and its normalized counterpart $Zp$) \cite{Dunn:2008}, $Zpx$ and $\Delta Zp$ \cite{Dickson:2010p237}. A coevolution network file is also produced which can be visualized using Graphviz \cite{Ellson:2002wf}.
\section*{Conclusions}
It is established that $MI$ by itself is not particularly accurate in predicting coevolving positions because it correlates with Entropy \cite{Martin:2005}, misleading phylogenetic signal \cite{Dunn:2008}, and alignment errors \cite{Dickson:2010p237}. Furthermore, analyzing gaps as the "21st amino acid" causes misleading results which is partially why the aforementioned studies excluded positions containing gaps from the analysis (Dickson et al. submitted). It is possible to overcome some limitations of raw $MI$ by using various corrections to $MI$ \cite{Dunn:2008,Little:2009,Buslje:2009ct,Dickson:2010p237}. Typically these corrections based on an analysis of raw $MI$ values are computationally inexpensive and so heavy optimization is not necessary. Thus the algorithm and software described herein can be used to reduce the time and memory required to calculate most $MI$-derived statistics.
The MIpToolset has been tested on Unix-like operating systems and is implemented in C for efficiency, with a Perl wrapper for handling input/output issues. The speed and efficiency of the MIpToolset has allowed for efficient database-wide analysis \cite{Dickson:2010p237} and the detection of protein family misalignments using an $MI$-derived method in real-time as the user edits their alignment in the software tool LoCo \cite{Dickson:2012jx}. To our knowledge, this is the fastest implementation of the coevolution statistics $MIp$, $Zp$, $Zpx$, and $\Delta Zp$\cite{Dunn:2008, Dickson:2010p237}.
It is available at: https://sourceforge.net/projects/miptoolset/
\section*{Authors contributions}
RJD designed the algorithm, and wrote the manuscript. GBG designed the project. All authors contributed to the software, and read and approved the final manuscript.
\section*{Acknowledgements}
The authors wish to thank the students of the 2011 and 2012 Biochemistry 4445a class at UWO for identifying documentation ambiguities and cross-platform issues. Also, thank you to Stan Dunn, Lindi Wahl, David Edgell, Thomas McMurrough, Jean Macklaim, Andrew Fernandes, Ardeshir Goliaei for helpful discussions and testing. Finally thank you to existing users of the MIpToolset for their feedback while using this software while it was in development.
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