text stringlengths 128 2.05k |
|---|
[EQUATION] vanish (or both) for every such [MATH] Since [MATH] where [MATH] we would get the absurd conclusion that [EQUATION] Equivalently, |
[EQUATION] This is not possible and the contradiction ends the proof. 4.6. Multiplicity of the spectrum of [MATH] We begin by observing that [MATH] and that [MATH] . Then any real eigenfunction which is a linear combination of [MATH] and [MATH] is |
[EQUATION] for some constant [MATH] . In local coordinates, [MATH] is [MATH] , and therefore we see that [EQUATION] For [MATH] and [MATH] such that [MATH] , we argue that [MATH] is satisfied for an open dense subset of metric [MATH] . This immediately implies the first, third, and the fourth statement of Theorem 1.3 . ... |
{lemm} Let [MATH] be a non-trivial principal [MATH] bundle. Fix integers [MATH] and [MATH] such that [MATH] . Among all [MATH] -invariant metric [MATH] on [MATH] [MATH] satisfying [MATH] is dense. |
Proof 4.4 The deformation of the base of the Kaluza-Klein metric does not touch the vertical operator [MATH] and therefore the first order perturbation equations for infinitesimal deformations of the base metric [MATH] gives, |
[EQUATION] Taking the inner product with [MATH] gives [EQUATION] If there exist weights [MATH] for which we cannot split the eigenvalue |
[MATH] then for all infinitesimal base perturbations [MATH] we get [EQUATION] Write [MATH] . Differentiation of the eigenvalue equation therefore gives the well-known formula |
[EQUATION] for every variation of [MATH] , where the inner product is that of [MATH] Recall from previous section that [MATH] . Because [MATH] we have for any [MATH] |
[EQUATION] Thus, [EQUATION] Integrating both sides against [MATH] and using that both eigenfunctions are [MATH] normalized gives |
[EQUATION] i.e., [MATH] 5. Local structure of eigensections at zeros To study the nodal sets of real and imaginary parts of Kaluza-Klein Laplacians, we first study the zeros of the associated sections of the line bundles. For simplicity of exposition, we assume that [MATH] and describe the zero sets of eigen- [MATH] -d... |
We follow the notation and terminology in the theory of holomorphic quadratic differentials, even though our eigendifferentials are [MATH] , usually not holomorphic and of general weight [MATH] Following a standard terminology for quadratic differentials, we call a point [MATH] such that [MATH] a “regular point” and a ... |
After the first version of this article was written, we located some recent articles generalizing the geometric properties of quadratic differentials on Riemann surfaces to [MATH] higher order differentials |
and to other line bundles. We now use the terminology and results of these articles but have retained some from our first version since it is important for us to lift to [MATH] |
5.1. Trajectories of eigen-differentials. The real and imaginary parts of the eigendifferentials [MATH] are called binary differentials of degree [MATH] |
and the equation for the zero set of [MATH] is called a binary differential equation of degree [MATH] . It is traditional to consider the nodal set [MATH] . If there exist exactly [MATH] |
solutions at a regular point where [MATH] then [MATH] is called totally real in Our [MATH] -differentials are of a special type since they are real and imaginary parts of [MATH] and therefore only have terms of the form [MATH] or [MATH] The following is the key input into Proposition |
{lemm} [MATH] is a totally real [MATH] -differential. At a regular point [MATH] , there exist [MATH] distinct solutions [MATH] of [MATH] in [MATH] |
Proof 5.1 It f [MATH] then in the notation of ( 1.4 ), the equation is [EQUATION] Here [MATH] and the equation is [EQUATION] where [MATH] and where we assume with no loss of generality that [MATH] Since the principal branch of [MATH] is one-to-one, there exists precisely one solution |
[MATH] of [MATH] with [MATH] , namely the principal branch of [MATH] Since [MATH] is [MATH] -periodic, [MATH] is [MATH] -periodic, and the full set of solutions is [MATH] with [MATH] |
The kernel of [MATH] defines a smooth [MATH] -valued distribution on [MATH] with singularities where [MATH] The [MATH] line fields defines a web of m transverse singular foliations, whose leaves are called the trajectories |
{defi} The trajectories of the [MATH] differential [MATH] are the integral curves of the kernel of [MATH] , i.e. the trajectories are the (smooth) curves [MATH] in [MATH] along which [MATH] |
{rema} A trajectory in this sense of this article is called a ‘horizontal trajectory’ in 19 , Definition 5.5.3] They are illustrated in |
19 , Section 7] for holomorphic quadratic differentials. Illustrations of webs for higher order real differentials can be found in |
Trajectories downstairs on [MATH] lift to [MATH] by their tangent vectors. A trajectory [MATH] downstairs is a smooth curve along which |
[EQUATION] It lifts to a smooth curve [MATH] in the nodal set upstairs. Since [MATH] is an isomorphism, the trajectories are special curves on the nodal set [MATH] |
5.2. Non-degenerate singular points The structure of the trajectories through a singular (zero) may be complicated in general if no conditions are placed on the degeneracy of the zeros. The purpose of Theorem is to allow us to assume that the zeros are of first order, so that they are isolated and non-degenerate. |
The structure of the trajectories of a totally real [MATH] -differential near an isolated singular point is discussed in As with vector fields, the key topological invariant of the singular point is its index |
{defi} The index of a singular point [MATH] where [EQUATION] is related to the degree of the circle map defined by [EQUATION] on a small circle around [MATH] to [MATH] by |
[EQUATION] Equivalently, in a small circle [MATH] around [MATH] , choose a unit vector [MATH] where [MATH] is a constant speed parametrization of [MATH] and let [MATH] be its length. Let [MATH] be a smooth extension of [MATH] along [MATH] After a complete turn, [MATH] must be one of the [MATH] solutions of [MATH] . Aft... |
be a smooth determination of the angle between the tangent line to [MATH] and [MATH] . Then [MATH] and [MATH] differ by an integer multiple of [MATH] . The index of [MATH] is defined by |
[EQUATION] Thus, the index has the form [MATH] with [MATH] The following Lemma shows that singular points must exist when the genus of [MATH] is non-zero. |
{lemm} If [MATH] has isolated non-degenerate zeros, then the sum of the indices of the zeros is the Chern class of [MATH] {lemm} |
If [MATH] is a non-degenerate singular point (zero of order [MATH] ) of [MATH] then [MATH] Proof 5.2 This follows from the fact that [MATH] is linear in this case and hence the degree of the associated circle map is [MATH] |
Proposition 12 For a generic Riemannian metric [MATH] on [MATH] , all singular points of all eigendifferentials of [MATH] on [MATH] have index [MATH] for all [MATH] |
Proof 5.3 It is part of Theorem , all singular points are non-degenerate. To prove this it suffices to show that the coefficients [MATH] are linear near each singular point. This follows from the Bers local formula for eigensections around a zero. We use Proposition 3.3 to Taylor expand the operator |
[EQUATION] around a nodal point. Let [MATH] be a nodal point of [MATH] . We Taylor expand the coefficients in Kähler normal coordinates for [MATH] |
in a disc [MATH] to get [MATH] [MATH] Thus, the osculating constant coefficient operator is [EQUATION] Let [MATH] denote homogeneous polynomials of degree [MATH] in [MATH] . It is better to arrange the terms of the Taylor expansion of [MATH] at [MATH] into terms |
[EQUATION] where [MATH] Thus, [MATH] [MATH] [MATH] etc. Note that [MATH] because [MATH] and [MATH] , so neither the second or first derivative terms contribute at this order. |
Also expand [EQUATION] where [MATH] is homogeneous of order [MATH] The following is the generalization of the Bers local expansion theorem to complex line bundles. |
{lemm} Let [MATH] be a zero of [MATH] . The first non-zero homogeneous term [MATH] of the Taylor expansion of an eigenfunction is a harmonic homogeneous polynomial. If the order of vanishing is [MATH] [MATH] In particular, at a non-degenerate zero, the first homogeneous term is [MATH] |
Proof 5.4 It is evident that [MATH] . If [MATH] is the term of lowest degree in the expansion of [MATH] then [MATH] , i.e., [MATH] is a homogeneous harmonic polynomial. In real dimension [MATH] the only possibilities are linear combinations of the real and imaginary parts of [MATH] . By a well-known argument, the nodal... |
This completes the proof of the Proposition. 6. Adapted Kaluza-Klein metrics All of the Kaluza-Klein metrics are Riemannian metrics on principal [MATH] bundles |
[MATH] associated to [MATH] complex Hermitian line bundles [MATH] . Given [MATH] we recover [MATH] as an associated line bundle. Let [MATH] be any Riemannian surface. We denote the genus of [MATH] by [MATH] . Let [MATH] be any complex line bundle with Hermitian metric [MATH] . Associated to [MATH] |
is the [MATH] bundle [MATH] of orthonormal frames. Let [MATH] generate the [MATH] action. We endow [MATH] with a connection [MATH] , that is, an [MATH] invariant [MATH] -form on [MATH] such that [MATH] |
The connection defines a splitting [EQUATION] into horizontal and vertical spaces. The vertical space is given by orbits of the [MATH] action. The horizontal space is defined by [MATH] and is isomorphic under [MATH] to [MATH] where [MATH] |
{defi} The Kaluza-Klein metric on [MATH] is the [MATH] -invariant metric [MATH] such that the horizontal space [MATH] is isometric to [MATH] , so that [MATH] is orthogonal to [MATH] and is invariant under the natural [MATH] action and so that the fiber is a unit speed geodesic. |
A Kaluza-Klein metric on the principal [MATH] bundle [MATH] is thus determined by the pair [MATH] where [MATH] is a metric on [MATH] and where [MATH] is a connection [MATH] -form on [MATH] . In general, the metric and connection are chosen independently. In Section 2.6 we discuss the orthonormal frame bundle, where |
[MATH] is the Riemannian connection of [MATH] Given [MATH] and any character [MATH] of [MATH] we obtain associated line bundles (resp, real rank 2 bundles) by |
[EQUATION] For purposes of this paper it may be assumed that [MATH] We often assume that [MATH] is equipped with a complex structure [MATH] and that [MATH] is a holomorphic line bundle. Let [MATH] be the unit co-disc bundle with respect to [MATH] and let |
[MATH] be its boundary, an [MATH] bundle [MATH] {rema} Not all [MATH] invariant metrics on [MATH] are adapted Kaluza-Klein metrics. It would be interesting to consider more general [MATH] -invariant metrics on [MATH] or on other manifolds (of all dimensions), as well as invariant metrics under more general compact Lie ... |
6.1. Geometry and analysis of Kaluza-Klein metrics We use the term Kaluza-Klein metric or Kaluza-Klein metric in the sense of Definition |
to denote metrics [MATH] on the unit tangent bundles [MATH] over surfaces for which the vertical and horizontal spaces are orthogonal and which is invariant under the free [MATH] |
action . They are special cases of Riemannian submersions with totally geodesic fibers isometric to [MATH] {defi} If [MATH] is a compact Lie group, and if [MATH] is a principal [MATH] -bundle with fiber [MATH] then one says that a connection [MATH] is an [MATH] -connection for [MATH] if the [MATH] action preserves the ... |
is adapted to [MATH] if the fibers [MATH] are totally geodesic and isometric to [MATH] and such that the horizontal distribution of [MATH] is the orthogonal complement to the vertical. |
The following Lemma gives details on the equivalences of the various conditions and is implicitly contained in , Example 2.1] and is proved in 21 , Theorem 3.5] . Hence we only sketch the proof. |
{lemm} Suppose that [MATH] acts freely on [MATH] and that [MATH] is an [MATH] invariant metric for which all orbits are geodesics isometric to [MATH] . Then [MATH] is a Kaluza-Klein metric and [MATH] is a Riemannian submersion with totally geodesic fibers isometric to [MATH] |
Proof 6.1 Under the freeness assumption, we have an [MATH] bundle [MATH] Let [MATH] be the vertical space, i.e., the tangent space to the orbits. Let [MATH] The metric [MATH] determines a quotient Riemannian metric on [MATH] using the isomorphisms |
[MATH] By assumption, [MATH] if we identify [MATH] . The only non-trivial statement is that the orbits are geodesics. Let [MATH] |
denote the orbit of a point [MATH] under the [MATH] action. Let [MATH] be a horizontal vector at [MATH] . Let [MATH] be the parallel translation of [MATH] along a curve in [MATH] with initial tangent vector [MATH] . Then [MATH] |
is a horizontal vector field along [MATH] . The arclength of the curve [MATH] is constant in [MATH] . The first variation formula implies that [MATH] is geodesic. |
These adapted metrics are special cases of invariant metrics on an [MATH] -manifold [MATH] . For instance, in the case of the non-free action of [MATH] on the standard [MATH] by rotations around the [MATH] -axis, |
[MATH] varies with the orbit, and almost no orbits are geodesics. It would be interesting to consider generalizations to all [MATH] invariant metrics. |
Sections of [MATH] , i.e., differentials of type [MATH] , lift to the dual line bundle [MATH] as equivariant scalar functions [MATH] transforming by [MATH] under the [MATH] action of rotating a frame. Using the metric [MATH] we form the unit tangent bundle [MATH] . In this section, we review the relevant formulae for l... |
6.2. Lifts to [MATH] The natural inner product on [MATH] is given by [EQUATION] Sections [MATH] of [MATH] naturally lift to [MATH] and [MATH] by |
[EQUATION] It is straightforward to check that the lift of [MATH] satisfies [MATH] and that [EQUATION] Indeed, if [MATH] then [MATH] |
In the case of [MATH] the lift has the form, [EQUATION] We define a orthonormal frame of [MATH] by [MATH] as above, and let [MATH] be the dual frame. In local coordinates |
[MATH] on [MATH] and in this local frame we define local coordinates [MATH] on [MATH] corresponding to the point [MATH] Then [MATH] lifts to the function, |
[EQUATION] Consequently, the eigendifferential [MATH] lifts to [EQUATION] In ( 1.2 ) we decomposed the lift into real and imaginary parts. We now relate them to the real and imaginary parts of [MATH] |
If we take the inner product of [MATH] and [MATH] just along the fiber and use orthogonality of [MATH] and that [MATH] , and then integrate in [MATH] |
we get {lemm} [MATH] 6.3. Eigenspace decompositions The Kaluza-Klein Laplacian has the form [EQUATION] is the horizontal Laplacian. The fact that the fiber Laplacian is [MATH] reflects the fact that [MATH] orbits are geodesics isometric to [MATH] |
The weight spaces are [MATH] -invariant, i.e., as an unbounded self-adjoint operator, [EQUATION] Under the canonical identification |
[EQUATION] using the lifting map and [MATH] under the lifting map. We then consider joint eigenfunctions [MATH] of the Kaluza-Klein Laplacian [MATH] and of [MATH] The commutation relations show that [MATH] |
{lemm} The Bochner Laplacian agrees with the horizontal Laplacian [MATH] . In the above local coordinates and frame, [EQUATION] Note that except for the last identity, these statements are true for any isometric [MATH] action, not just for adapted Kaluza-Klein metrics. |
6.4. Equivariant decomposition Since [MATH] acts isometrically on [MATH] we may decompose into its weight spaces, [EQUATION] where |
[EQUATION] The weight spaces are [MATH] -invariant, i.e., as an unbounded self-adjoint operator, [EQUATION] The lifting map gives a canonical identification |
[EQUATION] 7. Connectivity of nodal sets of Kaluza-Klein eigenfunctions Given the preparations in Section , it is now a simple matter to prove Theorem 1.4 The following is an immediate consequence of Lemma 5.1 |
{lemm} If [MATH] is a regular value, then [MATH] is a singular [MATH] -fold cover of [MATH] with blow-down singularities over points where [MATH] |
Indeed, the [MATH] zeros of [MATH] in [MATH] give [MATH] points on the fiber [MATH] in [MATH] . Since locally there exist [MATH] smooth determinations of the zeros, the nodal set is a covering map away from the singular points. |
We have separated Lemma from further geometric results on the map [MATH] in the next section since it was stated separately from those results in Theorem 1.4 |
{rema} In the literature, [MATH] is sometimes called a branched cover , Section 4] , but as J. Y. Welschinger explained to us, the terminology is misleading since smooth branched covers are supposed to have [MATH] singularities over the branch points, just as for holomorphic branched covers, while the inverse image of ... |
8. Nodal domains of real and imaginary parts We now give a sketch of the proof of Theorem 1.3 By Proposition (and ( 1.5 )), [MATH] |
is a [MATH] -sheeted cover. Moreover, [MATH] is an [MATH] bundle and [EQUATION] is a fiber bundle whose fibers consist of the punctured fibers [MATH] . The connected components of each punctured fiber consist of ‘arcs’ along which [MATH] has a constant sign. We therefore express it as |
[EQUATION] where [MATH] in [MATH] . Each [MATH] is a fiber bundle whose fiber consists of [MATH] arcs of the fibers of [MATH] . Since the number of zeros in each regular fiber is [MATH] , the number of connected components of [MATH] is [MATH] . When we take the closure of these sets (i.e., add in the singular fibers, o... |
We now use the local analysis in Section of eigendifferentials of generic Bochner Laplacians around their zeros to determine how the sheets are connected at the singular fibers [MATH] corresponding to singular points (i.e., zeros) of [MATH] i.e., we consider the maximal components [MATH] of |
[EQUATION] in which [MATH] has a single sign. When we union the left side with [MATH] we glue together some of these domains along intervals of the singular fibers. |
The gluing rule for the nodal domains is determined by the gluing rule for the nodal set, since the boundary of the the each nodal domain is the nodal set. From the downstairs point of view, the gluing rule is the monodromy of the cover [MATH] If we fix a singular point [MATH] , then we get a monodromy representation |
[EQUATION] determining how the sheets of the nodal set are changed as the point circles around [MATH] By Proposition 12 , the index of the singular points [MATH] is [MATH] . In terms of the monodromy, this means precisely that each turn around a circle [MATH] enclosing [MATH] lifts to an arc from one vector in the fibe... |
It follows that both the [MATH] region and [MATH] region is connected in [MATH] . Hence there are just two nodal domains. 8.1. Counting the number of nodal domains |
We now give a more detailed presentation. Let [MATH] be an open disc. We first study connectivity of a certain graph that arise from a pair of partitions of [MATH] |
Let [MATH] and [MATH] be partitions of [MATH] , i.e., [MATH] (resp. [MATH] ) is a collection of disjoint open-sets [MATH] (resp. [MATH] ) such that |
[EQUATION] Let [MATH] and [MATH] be colorings of [MATH] and [MATH] , and define the inversions of [MATH] and [MATH] by [MATH] and [MATH] |
We now define a graph [MATH] as follows: The vertex set is [EQUATION] and edges are [EQUATION] for [MATH] with the identification [MATH] |
{defi} We say a pair of partitions [MATH] generic, if [EQUATION] does not contain a closed curve. {lemm} For a generic pair of partitions [MATH] with any given colorings [MATH] and [MATH] , any connected component of [MATH] contains at least one of the following [MATH] vertices: |
[EQUATION] In particular, [MATH] has at most [MATH] connected components. Proof 8.1 We first consider the case [MATH] . To claim [MATH] has only [MATH] connected components, it is sufficient to prove that if [MATH] , then [MATH] and [MATH] are path-connected. |
Because [MATH] is a generic pair, one can find a chain of open-sets [EQUATION] such that two adjacent open-sets have non-trivial intersection. |
Observe that if [MATH] , then either [EQUATION] is an edge, and likewise either [EQUATION] is an edge. Therefore the above chain of open-sets corresponds to a path connecting [MATH] with either [MATH] or [MATH] . However, from the assumption [MATH] , and from the construction of [MATH] [MATH] cannot be connected to [MA... |
Now for the rest, note that [MATH] is an [MATH] -covering of [MATH] , and because [MATH] and [MATH] belongs to the different connected components of [MATH] , any connected components of [MATH] must contain at least one vertex of the fiber of [MATH] or [MATH] |
For a large class of colorings, we can deduce a much stronger result. {lemm} Let [MATH] be a generic pair of partitions. Assume that we are given with a pair of colorings [MATH] and [MATH] |
There exist four open sets [MATH] such that [EQUATION] for [MATH] and [MATH] , and that [MATH] Then the graph [MATH] has [MATH] connected components. |
Proof 8.2 Note that any connected component of [MATH] must contain either one of [MATH] or one of [MATH] with [MATH] , because [MATH] has only two connected components. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.