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Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If $\pi$ is a spacelike 2 plane, let $R(\pi)$ be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) has constant rank 2 and show these are geometrically realizable by hypersurfaces in flat spaces. We also classify the Ivanov-Petrova algebraic curvature tensors of rank 2; these are the algebraic curvature tensors of constant rank 2 such that the complex Jordan normal form of R(-) is constant. | Algebraic curvature tensors whose skew-symmetric curvature operator has
constant rank 2 | 14,600 |
We construct a family of pseudo-Riemannian manifolds so that the skew-symmetric curvature operator, the Jacobi operator, and the Szabo operator have constant eigenvalues on their domains of definition. This provides new and non-trivial examples of Osserman, Szabo, and IP manifolds. We also study when the associated Jordan normal form of these operators is constant. | Szabo Osserman IP Pseudo-Riemannian manifolds | 14,601 |
A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in euclidean 3-space with a point of positive curvature and no skew loops are the quadrics. In particular, ellipsoids are the only closed surfaces without skew loops. We also prove results about skew loops on cylinders and positively curved surfaces. | Skew loops and quadric surfaces | 14,602 |
We show that the natural S^1-bundle over a projective special Kaehler manifold carries the geometry of a proper affine hypersphere endowed with a Sasakian structure. The construction generalizes the geometry of the Hopf-fibration $\Sr^{2n+1} \longrightarrow \CP^n$ in the context of projective special Kaehler manifolds. As an application we have that a natural circle bundle over the Kuranishi moduli space of a Calabi-Yau threefold is a Lorentzian proper affine hypersphere. | Proper Affine Hyperspheres which fiber over Projective Special Kaehler
Manifolds | 14,603 |
We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with three regular ends which are asymptotic to catenoid cousins (CMC-1 trinoids). | Hyperbolic constant mean curvature one surfaces: Spinor representation
and trinoids in hypergeometric functions | 14,604 |
Following the point of view of Gray and Hervella, we derive detailed conditions which characterize each one of the classes of almost quaternion-Hermitian $4n$-manifolds, $n>1$. Previously, by completing a basic result of A. Swann, we give explicit descriptions of the tensors contained in the space of covariant derivatives of the fundamental form $\Omega$ and split the coderivative of $\Omega$ into its ${\sl Sp}(n){\sl Sp}(1)$-components. For $4n>8$, A. Swann also proved that all the information about the intrinsic torsion $\nabla \Omega$ is contained in the exterior derivative ${\sl d} \Omega$. Thus, we give alternative conditions, expressed in terms of ${\sl d} \Omega$, to characterize the different classes of almost quaternion-Hermitian manifolds. | Almost Quaternion-Hermitian Manifolds | 14,605 |
Conformal Killing forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We show the existence of conformal Killing forms on nearly Kaehler and weak G_2-manifolds. Moreover, we give a complete description of special conformal Killing forms. A further result is a sharp upper bound on the dimension of the space of conformal Killing forms. | Conformal Killing forms on Riemannian manifolds | 14,606 |
The aim of this article is to proof a necessary and sufficient condition for the existence of a Cartan connection on a principal bundle. After collecting the essentially well known facts to fix the terminology, soldering forms and geometrizable principal bundles are defined to finally prove the existence criterion. | The Existence of Cartan Connections and Geometrizable Principal Bundles | 14,607 |
In this paper we relate the Fefferman-Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tractor bundle and the normal standard tractor connection, which are equivalent to the Cartan bundle and the Cartan connection. This result is applied to obtain a procedure to get tractor formulae for all conformal invariants that can be obtained from the ambient metric construction. We also get information on ambient metrics which are Ricci flat to higher order than guaranteed by the results of Fefferman-Graham. | Standard Tractors and the Conformal Ambient Metric Construction | 14,608 |
In this note we show that the property of having only vanishing triple Massey products in the equivariant cohomology is inherited by the set of fixed points of hamiltonian circle actions on closed symplectic manifolds. This result can be considered in a more general context of characterizing homotopic properties of Lie group actions. In particular, it can be viewed as a partial answer to the Allday-Puppe question about finding conditions ensuring the "formality" of G-actions. | A note on hamiltonian Lie group actions and Massey products | 14,609 |
A Sim(n-1,1) affine manifold is an affine manifold whose linear holonomy is contained in the similarity lorentzian group but not in the lorentzian group. The class of similarity lorentzian affine manifolds is a small part in the nice class of conformally lorentzian flat manifolds. In this paper we show that a compact Sim(n-1,1) affine manifold is incomplete. We characterize the universal cover of radiant compact Sim(n-1,1) affine manifolds whose developing map is injective. Let q be the quadratic form which define the Sim(n-1,1) structure, using riemannian foliation theory, we classify compact radiant Sim(n-1,1) affine manifolds M such that D(M') the image of the universal cover M' of M by the developing map D is contained in the upper cone defined q | Closed similarity lorentzian affine manifolds | 14,610 |
We give a topological interpretation of the space of L2-harmonic forms on finite-volume manifolds with sufficiently pinched negative curvature. We give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants. The method consists first in comparing L2-cohomology with weighted L2-cohomology thanks to previuos works done by T. Ohsawa, and then in identifying these weighted spaces. | Sur la L2-cohomologie des varietes a courbure negative | 14,611 |
Given a compact Riemannian manifold together with a group of isometries, we discuss MCF of the orbits and some applications: eg, finding minimal orbits. We then specialize to Lagrangian orbits in Kaehler manifolds. In particular, in the Kaehler-Einstein case we find a relation between MCF and moment maps which, for example, proves that the minimal Lagrangian orbits are isolated. | Mean Curvature Flow, Orbits, Moment Maps | 14,612 |
Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let $P = P_{g,k} = r_{g,k} \times R_+^k$ be the space of parabolic structures over Riemann surfaces of genus g with k (marked) punctures, the real analytic structure coming from the 3g-3+k local complex analytic coordinates on the Riemann moduli space r_{g,k}. Then the parabolic classifying map, Phi: M --> P, which assigns to a CMC surface its induced conformal structure and asymptotic necksizes, is a proper, real analytic map. It follows that Phi is closed and in particular has closed image. For genus g=0, this can be used to show that every conformal type of multiply punctured Riemann sphere occurs as a CMC surface, and -- under a nondegeneracy hypothesis -- that Phi has a well defined (mod 2) degree. This degree vanishes, so generically an even number of CMC surfaces realize any given conformal structure and asymptotic necksizes. | Conformal Structures and Necksizes of Embedded Constant Mean Curvature
Surfaces | 14,613 |
All complete, axially symmetric surfaces of constant mean curvature in R^3 lie in the one-parameter family D_tau of Delaunay surfaces. The elements of this family which are embedded are called unduloids; all other elements, which correspond to parameter value tau element in R^-, are immersed and are called nodoids. The unduloids are stable in the sense that the only global constant mean curvature deformations of them are to other elements of this Delaunay family. We prove here that this same property is true for nodoids only when tau is sufficiently close to zero (this corresponds to these surfaces having small `necksizes'). On the other hand, we show that as tau decreases to negative infinity, infinitely many new families of complete, cylindrically bounded constant mean curvature surfaces bifurcate from this Delaunay family. The surfaces in these branches have only a discrete symmetry group. | Bifurcating nodoids | 14,614 |
This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmueller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the Thurston classification. The second result concerns the asymptotic behavior of these geodesics. It is shown that geodesics that are equivariant with respect to independent pseudo-Anosov's diverge. It follows that subgroups of the mapping class group which contain independent pseudo-Anosov's act in a reductive manner with respect to the Weil-Petersson geometry. This implies an existence theorem for equivariant harmonic maps to the metric completion. | Classification of Weil-Petersson Isometries | 14,615 |
Real flag manifolds are the isotropy orbits of noncompact symmetric spaces $G/K$. Any such manifold $M$ enjoys two very peculiar geometric properties: It carries a transitive action of the (noncompact) Lie group $G$, and it is embedded in euclidean space as a taut submanifold. The aim of the paper is to link these two properties by showing that the gradient flow of any height function is a one-parameter subgroup of $G$, where the gradient is defined with respect to a suitable homogeneous metric $s$ on $M$; this generalizes the Kaehler metric on adjoint orbits (the so-called complex flag manifolds). | Steepest descent on real flag manifolds | 14,616 |
Let $(M,F)$ be a Finsler manifold. We construct a 1-cocycle on $\Diff(M)$ with values in the space of differential operators acting on sections of some bundles, by means of the Finsler function $F.$ As an operator, it has several expressions: in terms of the Chern, Berwald, Cartan or Hashiguchi connection, although its cohomology class does not depend on them. This cocycle is closely related to the conformal Schwarzian derivatives introduced in our previous work. The second main result of this paper is to discuss some properties of the conformally invariant quantization map by means of a Sazaki (type) metric on the slit bundle $TM\backslash 0$ induced by $F.$ | Remarks on the Schwarzian Derivatives and the Invariant Quantization by
means of a Finsler Function | 14,617 |
In this article we prove an upper bound for a Hilbert polynomial on quaternionic Kaehler manifolds of positive scalar curvature. As corollaries we obtain bounds on the quaternionic volume and the degree of the associated twistor space. Moreover the article contains some details on differential equations of finite type. Part of it is used in the proof of the main theorem. | An upper bound for a Hilbert polynomial on quaternionic Kahler manifolds | 14,618 |
We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity k representations, we classify the irreducible representations of copolarity one, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Samelson. | Copolarity of isometric actions | 14,619 |
Generalizing results due to Brady and Farb we prove the existence of a bilipschitz embedded manifold of pinched negative curvature and dimension m_1+m_2-1 in the product X:=X_1^{m_1} times X_2^{m_2} of two Hadamard manifolds X_i^{m_i} of dimension m_i with pinched negative curvature. Combining this result with a Theorem by Buyalo and Schroeder we prove the additivity of the hyperbolic rank for products of manifolds with pinched negative curvature. | Hyperbolic Rank of Products | 14,620 |
We prove that for a mean curvature flow of a compact symplectic surface in a compact Kaehler-Einstein surface, the tangent cone at the first blow-up time consists of a finite union of more than two 2-planes in $R^4$ which are complex in a complex structure on $R^4$. | Singularities of codimension two mean curvature flow of symplectic
surfaces | 14,621 |
The space of differential operators acting on skewsymmetric tensor fields or on smooth forms of a smooth manifold are representations of its Lie algebra of vector fields. We compute the first cohomology spaces of these representations and show how they are related to the cohomology with coefficients in ther space of smooth functions of the manifold. | On the Cohomology of the spaces of differential operators acting on
skewsymmetric tensor fields or on forms, as modules of the Lie algebra of
vector fields | 14,622 |
Let Z be a smooth projective manifold. In these notes I will prove that the K-group of R-constructible sheaves is isomorphic to the free abelian group with one generator for each open semialgebraic subset $U$ (which I will denote by the same letter) modulo the Mayer-Vietoris relations: U + V - U^V - UvV = 0. I will prove it by showing that both groups in question are isomorphic to the group of all integer-valued semialgebraic functions on Z. | The K-group of R-constructible Sheaves | 14,623 |
We investigate the spectra of a family of pairs (M_i,A_i) consisting of a complete Riemannian manifold M_i and a closed subset A_i and which converge in the Lipschitz topology to a pair (M,A). This is used to construct manifolds of bounded curvature, nonempty essential spectrum, infinitely many eigenvalues below the essential spectrum and with an arbitrary number of such eigenvlues of arbitrarily high multiplicity. | Spectral convergence of manifold pairs | 14,624 |
We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of the classical dynamical Yang-Baxter equation on simple Lie algebras as classified by the same authors. Our approach is based on the study of a class of Poisson structures on trivial Lie groupoids within the category of biequivariant Poisson manifolds. In the former case, it is shown that the dual Poisson groupoid of such a dynamical Poisson groupoid is isomorphic to a Poisson groupoid (with trivial Lie groupoid structure) within this category. In the latter case, we find that the dual Poisson groupoid is also of dynamical type modulo Poisson groupoid isomorphisms. For the coboundary dynamical Poisson groupoids associated with constant r- matrices, we give an explicit construction of the corresponding symplectic double groupoids. In this case, the symplectic leaves of the dynamical Poisson groupoid are shown to be the orbits of a Poisson Lie group action. | On Dynamical Poisson Groupoids I | 14,625 |
The projection of a compact oriented submanifold M^{n-1} in R^{n+1} on a hyperplane P^{n} can fail to bound any region in P. We call this ``projecting to zero.'' Example: The equatorial S^1 in S^2 projects to zero in any plane containing the x_3-axis. Using currents to make this precise, we show: A lipschitz (homology) (n-1)-sphere embedded in a compact, strictly convex hypersurface cannot project to zero on n+1 linearly independent hyperplanes in R^{n+1}. We also show, using examples, that all the hypotheses in this statement are sharp. | Projecting (n-1)-cycles to zero on hyperplanes in R^{n+1} | 14,626 |
For null curves in PSL(2,C), there exists a representation formula in terms of two meromorphic functions and their derivatives (Small's formula). In this paper, we give an elementary proof of Small's formula. Moreover, a similar formula for Legendrian curves in PSL(2,C) is given. As null curves in PSL(2,C) are related to mean curvature one surfaces in hyperbolic 3-space H^3, Legendrian curves are related to flat surfaces in H^3. So, as an application of Small-type formula for Lengendrian curves, we give new examples of flat surfaces in H^3. | An elementary proof of Small's formula for null curves in PSL(2,C) and
an analogue for Legendrian curves in PSL(2,C) | 14,627 |
This paper classifies Hermitian structures on 6-dimensional nilmanifolds M=G/L for which the fundamental 2-form is d d-bar closed, a condition that is shown to depend only on the underlying complex structure J of M. The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obtained from a limacon-shaped curve in the complex plane. Related theory provides examples of various types of Ricci-flat structures. | Families of strong KT structures in six dimensions | 14,628 |
We propose a method of constructing completely integrable systems based on reduction of bihamiltonian structures. More precisely, we give an easily checkable necessary and sufficient conditions for the micro-kroneckerity of the reduction (performed with respect to a special type action of a Lie group) of micro-Jordan bihamiltonian structures whose Nijenhuis tensor has constant eigenvalues. The method is applied to the diagonal action of a Lie group $G$ on a direct product of $N$ coadjoint orbits $\O=O_1\times...\times O_N$ endowed with a bihamiltonian structure whose first generator is the standard symplectic form on $\O$. As a result we get the so called classical Gaudin system on $\O$. The method works for a wide class of Lie algebras including the semisimple ones and for a large class of orbits including the generic ones and the semisimple ones. | Projections of Jordan bi-Poisson structures that are Kronecker, diagonal
actions, and the classical Gaudin systems | 14,629 |
Given a 1-parameter family of 1-forms $\g(t)= \g_0+t\g_1+...+t^n\g_n$, consider the condition $d\g(t)\wedge\g(t)=0$ (of integrability for the annihilated by $\g(t)$ distribution $w(t)$). We prove that in order that this condition is satisfied for any $t$ it is sufficient that it is satisfied for $N=n+3$ different values of $t$ (the corresponding implication for $N=2n+1$ is obvious). In fact we give a stronger result dealing with distributions of higher codimension. This result is related to the so-called Veronese webs and can be applied in the theory of bihamiltonian structures. | On integrability of generalized Veronese curves of distributions | 14,630 |
We investigate the linearizability problem for different classes of 4-webs in the plane. In particular, we apply a recently found in [AGL] the linearizability conditions for 4-webs in the plane to confirm that a 4-web MW (Mayrhofer's web) with equal curvature forms of its 3-subwebs and a nonconstant basic invariant is always linearizable (this result was first obtained in [M 28]); it also follows from the papers [Na 96] and [Na98]). Using the same conditions, we also prove that such a 4-web with a constant basic invariant (Nakai's web) is linearizable if and only if it is parallelizable. We also study four classes of the so-called almost parallelizable 4-webs APW_a, a = 1, 2, 3, 4 (for them the curvature K = 0 and the basic invariant is constant on the leaves of the web foliation X_a), and prove that a 4-web APW_a is linearizable if and only if it coincides with a 4-web MW_a of the corresponding special class of 4-webs MW. The existence theorems are proved for all the classes of 4-webs considered in the paper. | 4-webs in the plane and their linearizability | 14,631 |
We find d - 2 relative differential invariants for a d-web, d \geq 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f (x,y) and g_4 (x,y),...,g_d (x,y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g_4, and d - 4 PDEs of the second order with respect to f and g_4,...,g_d. For d = 4, this result confirms Blaschke's conjecture on the nature of conditions for the linearizabilty of a 4-web. We also give Mathematica codes for testing 4- and d-webs (d > 4) for linearizability and examples of their usage. | Linearizability of d-webs, d \geq 4, on two-dimensional manifolds | 14,632 |
The theory of complete surfaces of (nonzero) constant mean curvature in $\RR^3$ has progressed markedly in the last decade. This paper surveys a number of these developments in the setting of Alexandrov embedded surfaces; the focus is on gluing constructions and moduli space theory, and the analytic techniques on which these results depend. The last section contains some new results about smoothing the moduli space and about CMC surfaces in asymptotically Euclidean manifolds. | Recent advances in the global theory of constant mean curvature surfaces | 14,633 |
We study the Einstein-Dirac equation as well as the weak Killing equation on Riemannian spin manifolds with codimension one foliation. We prove that, for any manifold $M^n$ admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics $\bar{\eta}$ on $M^n \times {\mathbb R}$ such that $(M^n \times {\mathbb R}, \bar{\eta})$ admit Einstein spinors (resp. weak Killing spinors). To prove the result we split the Einstein-Dirac equation into evolution equations and constraints, by means of Cartan's frame formalism, and apply the local preservation property of constraints. | A Local Existence Theorem for the Einstein-Dirac Equation | 14,634 |
We obtain a maximum principle, and "a priori" upper estimates for solutions of a class of non linear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumption of volume growth conditions. Various applications of the results obtained are presented. | "A priori" estimates for solutions of singular elliptic inequalities and
applications | 14,635 |
For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary Clifford algebras, followed by quaternionic vectors as a special case. All results are shown to reduce to the established method of complexifying vector fields. For simplicity, differential forms are used rather than vector notation. | On hypercomplexifying real forms of arbitrary rank | 14,636 |
In this paper we introduce a generalisation of the notion of holonomy for connections over a bundle map on a principal fibre bundle. We prove that, as in the standard theory on principal connections, the holonomy groups are Lie subgroups of the structure group of the principle fibre bundle and we also derive a straightforward generalisation of the Reduction Theorem. | Leafwise holonomy of connections over a bundle map | 14,637 |
Any strictly pseudoconvex domain in C2 carries a complete Kahler-Einstein metric, the Cheng-Yau metric, with ``conformal infinity'' the CR structure of the boundary. It is well known that not all CR structures on the 3-sphere arise in this way. In this paper, we study CR structures on the 3-sphere satisfying a different filling condition: boundaries at infinity of (complete) selfdual Einstein metrics. We prove that (modulo contactomorphisms) they form an infinite dimensional manifold, transverse to the space of CR structures which are boundaries of complex domains (and therefore of Kahler-Einstein metrics). | Autodual Einstein versus Kahler-Einstein | 14,638 |
Let F_0=B,...,F_n be a sequence of differentiable manifolds, G_i a Lie subgroup of diffeomorphisms of F_i, and H_i a subgroup of G_i central in G_i. We suppose also given a locally trivial bundle p_{K_i} over F_{i-1} which typical fiber is K_i the quotient of G_i by H_i. The aim of this paper is to study the differential geometry of the following problem: classify sequences M_n\to...M_1, where each map from M_i to M_{i-1} is a locally trivial fibration which typical fiber is F_i and which transition functions image are elements of G_i. We associate to this problem a tower of gerbes and define for it the notion of connective structure, curvature and holonomy using the notion of free transitive distribution (free TD) | The diffential geometry of composition sequences of differentiable
manifolds | 14,639 |
We study the local structure of Lie bialgebroids at regular points. In particular, we classify all transitive Lie bialgebroids. In special cases, they are connected to classical dynamical $r$-matrices and matched pairs induced by Poisson group actions | The Local Structure of Lie Bialgebroids | 14,640 |
This paper studies rapidly forming singularities in the Yang-Mills flow. It is shown that a sequence of blow-ups near the singular point converges, modulo the gauge group, to a homothetically shrinking soliton with non-zero curvature. The proof uses Hamilton's monotonicity formula. Examples of homothetically shrinking solitons are given in the case of trivial bundles over R^n for dimensions 5 through 9. | Singularity formation in the Yang-Mills flow | 14,641 |
We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus of the complex projective plane and of the Whitney spheres in the complex projective, complex Euclidean and complex hyperbolic planes. | Lagrangian surfaces with circullar ellipse of curvature in complex space
forms | 14,642 |
We address the question: how large is the family of complete metrics with nonnegative sectional curvature on S^2xR^3? We classify the connection metrics, and give several examples of non-connection metrics. We provide evidence that the family is small by proving some rigidity results for metrics more general than connection metrics. | Rigidity for Nonnegatively Curved Metrics on S^2xR^3 | 14,643 |
This paper addresses Cheeger and Gromoll's question of which vector bundles admit a complete metric of nonnegative curvature, and relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle which admits a metric of nonnegative curvature must admit a connection, a tensor, and a metric on the base space which together satisfy a certain differential inequality. On the other hand, a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature. Our results sharpen and generalize Walschap and Strake's conditions under which a vector bundle admits a connection metric of nonnegative curvature. | Conditions for Nonnegative Curvature on Vector Bundles and Sphere
Bundles | 14,644 |
Let $x:M\to S^{n+p}$ be an $n$-dimensional submanifold in an $(n+p)$-dimensional unit sphere $S^{n+p}$, $x:M\to S^{n+p}$ is called a Willmore submanifold to the following Willmore functional: $$ \int_M(S-nH^2)^{\frac{n}{2}}dv, $$ where $S=\sum\limits_{\alpha,i,j}(h^\alpha_{ij})^2$ is the square of the length of the second fundamental form, $H$ is the mean curvature of $M$. In [13], author proved an integral inequality of Simon's type for $n$-dimensional compact Willmore hypersurfaces in $S^{n+1}$ and gave a characterization of {\it Willmore tori}. In this paper, we generalize this result to $n$-dimensional compact Willmore submanifolds in $S^{n+p}$. In fact, we obtain an integral inequality of Simon's type for compact Willmore submanifolds in $S^{n+p}$ and give a characterization of {\it willmore tori} and {\it Veronese surface} by use of integral inequality. | Willmore submanifolds in a sphere | 14,645 |
We define a `Higgs field' for a four-dimensional spin$^c$-manifold to be a smooth section of its positive half-spinor bundle, transverse to the zero section, and defined only up to a positive functional factor. This is intended to be a generalization of almost complex structures on real four-manifolds, each of which may in fact be treated as a Higgs field without zeros for a specific spin$^c$-structure. The notions of totally real or pseudoholomorphic immersions of real surfaces in an almost complex manifold of real dimension four have straighforward generalizations to the case of a spin$^c$-manifold with a Higgs field. Our results consist, first, in showing that totally real immersions of closed oriented surfaces in four-dimensional spin$^c$-manifolds with Higgs fields have, basically, the same properties as in the almost-complex case, and, secondly, in providing a description of all pseudoholomorphic immersions of such surfaces in the four-sphere endowed with a "standard" Higgs field. | Immersions of surfaces in spin$^c$-manifolds with Higgs fields | 14,646 |
We study affine immersions as introduced by Nomizu and Pinkall. We classify those affine immersions of a surface in 4-space which are degenerate and have vanishing cubic form (i.e. parallel second fundamental form). This completes the classification of parallel surfaces of which the first results were obtained in the beginning of this century by Blaschke and his collaborators. | Parallel surfaces in affine 4-space | 14,647 |
For a m-tuple a=(a_1,...,a_m) of positive real numbers, the robot arm of type a in R^d is the map f^a:(S^{d-1})^m -> R^d defined by f^a(z_1,...,z_m) to be the sum of the a_jz_j's. Our aim is to attack the inverse problem via the horizontal liftings for the distribution Delta^a orthogonal to the fibers of f^a. One shows that the connected components by horizontal curves are the orbits of an actiion on (S^{d-1})^m by a product of groups of Moebius transformations. In several cases, the holonomy orbits of the distribution Delta^a are also described. | Contrôle des bras articulés et transformations de Moebius (Control
of robot arms and Moebius transformations) | 14,648 |
We emphasize some properties of coherent state groups, i.e. groups whose quotient with the stationary groups, are manifolds which admit a holomorphic embedding in a projective Hilbert space. We determine the differential action of the generators of the representation of coherent state groups on the symmetric Fock space attached to the dual of the Hilbert space of the representation. This permits a realization by first-order differential operators with holomorphic polynomial coefficients on K\"ahler coherent state orbits. | Differential operators on orbits of coherent states | 14,649 |
The Bakry-Emery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the Bakry-Emery tensor. We show that the Bakry-Emery tensor is nondecreasing under a Riemannian submersion whose fiber transport preserves measures up to constants. We give some relations between the Bakry-Emery tensor and measured Gromov-Hausdorff limits. | Some Geometric Properties of the Bakry-Emery-Ricci Tensor | 14,650 |
We show that if $\nabla R$ is a Jordan Szabo algebraic covariant derivative curvature tensor on a vector space of signature (p,q), where q is odd and p is less than q or if q is congruent to 2 mod 4 and if p is less than q-1, then $\nabla R=0$. This algebraic result yields an elementary proof of the geometrical fact that any pointwise totally isotropic pseudo-Riemannian manifold with such a signature (p,q) is locally symmetric. | Jordan Szabo algebraic covariant derivative curvature tensors | 14,651 |
We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound. | The entropy formula for the Ricci flow and its geometric applications | 14,652 |
The main result of this paper is: Given any constant C, there is $(\epsilon,k,L)$ such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a $(\epsilon,k,L)$-neck, then the asymptotic scalar curvature ratio is bigger or equal to C. As a application we proved that the asymptotic scalar curvature ratio of a complete noncompact ancient Type I-like solution to the Ricci flow with bounded positive sectional curvature on an orientable 3-manifold, is infinity. | On the asymptotic scalar curvature ratio of complete Type I-like ancient
solutions to the Ricci flow on non-compact 3-manifolds | 14,653 |
In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent maximum principle and the other is a time-dependent maximum principle subject to an avoidance set. | The time-dependent maximum principle for systems of parabolic equations
subject to an avoidance set | 14,654 |
We prove an estimate for solutions to the linearized Ricci flow system on closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is preserved estimate for the Ricci curvatures of solutions to the Ricci flow on 3-manifolds with positive Ricci curvature. In our estimate we make no assumption on the curvature of the initial metric. We show that the norm of the solution of the Lichnerowicz Laplacian heat equation (coupled to the Ricci flow) is bounded by a constant (depending on time) times the scalar curvature plus a constant. This relies on a Bochner type formula and establishing the nonnegativity of a degree 4 homogeneous polynomial in 6 variables. | A pinching estimate for solutions of the linearized Ricci flow system on
3-manifolds | 14,655 |
Orbits of families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows for a global description of a smooth geometric structure on a family of manifolds in terms of a single object defined on the corresponding family of vector fields. Stratified spaces, Poisson spaces and almost complex spaces are discussed. | Orbits of families of vector fields on subcartesian spaces | 14,656 |
We prove a linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow. We then use this sharp differential inequality to study the Liouville properties of the plurisubharmonic functions on complete Kaehler manifolds with nonnegative bisectional curvature. | Plurisubharmonic functions and the Kaehler-Ricci flow | 14,657 |
We show that the solution constructed in an earlier work of Y-G. Shi and the authors can be used to obtain sharp gradient estimates for the Kaehler-Ricci flow which achieves equality on a steady soliton. The estimate can be applied to obtain a long time existence of the Kaehler-Ricci flow. In the second part of the paper we develope a so-called "moment" type estimate for the Kaehler-Ricci flow, through which we prove that the flow preserves several properties. In the last part we study the convergence of the Kaehler-Ricci flow. | Kaehler-Ricci flow and the Poincare-Lelong equation | 14,658 |
We give a new and complete proof of Hamilton's injectivity radius estimate for sequences with bounded and almost nonnegative curvature operators, unbounded diameters, and bump-like origins. Such sequences arise in particular from dilations about a singularity of the Ricci flow on a 3-manifold. | Hamilton's injectivity radius estimate for sequences with almost
nonnegative curvature operators | 14,659 |
We obtain a lower bound for the diameter of a solution to the Ricci flow on a compact manifold with nonvanishing first real cohomology. A consequence of our result is an affirmative answer to Hamilton's conjecture that a product metric on $(S^{1}\times S^{n-1}$ cannot arise as a final time limit flow. | A lower bound for the diameter of solutions to the Ricci flow with
nonzero $H^{1}(M^{n};R)$ | 14,660 |
We classify those curvature-homogeneous Einstein four-manifolds, of all metric signatures, which have a complex-diagonalizable curvature operator. They all turn out to be locally homogeneous. More precisely, any such manifold must be either locally symmetric or locally isometric to a suitable Lie group with a left-invariant metric. To show this we explicitly determine the possible local-isometry types of manifolds that have the properties named above, but are not locally symmetric. | Curvature-homogeneous indefinite Einstein metrics in dimension four: the
diagonalizable case | 14,661 |
We prove two geometric index theorems for a family of first-order elliptic operators over a manifold with boundary by computing eta form representatives for the Chern character classes of the index bundle. The eta forms occur as relative and regularized traces on infinite-dimensional vector bundles realized as the limiting values of superconnection character forms. | Eta forms and the Chern Character | 14,662 |
We prove that a complete noncompact K\"{a}hler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}$^{n}$ and we show that the manifold is topologically {\bf R}$^{2n}$. In particular, when $M^{n}$ is a K\"{a}hler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to {\bf C}$^{2}$. | On Complete Noncompact Kähler Manifolds with Positive Bisectional
Curvature | 14,663 |
In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional K\"ahler manifold $M$ of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have Euclidean volume growth and its scalar curvature decays to zero at infinity in the average sense, then $M$ is biholomorphic to $\C^2$. During the proof, we also discover an interesting gap phenomenon which says that a K\"ahler manifold as above automatically has quadratic curvature decay at infinity in the average sense. | A Uniformization Theorem Of Complete Noncompact Kähler Surfaces With
Positive Bisectional Curvature | 14,664 |
In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. We prove that a complete noncompact K\"ahler surface with positive and bounded sectional curvature and with finite analytic Chern number $c_{1}(M)^{2}$ is biholomorphic to ${\C}^2$. | Positively Curved Complete Noncompact Kähler Manifolds | 14,665 |
In this paper we obtain three results concerning the geometry of complete noncompact positively curved K\"{a}hler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact K\"{a}hler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex dimension). The second one states that the curvature of a complete noncompact K\"{a}hler manifold with positive bisectional curvature decays at least linearly in the average sense. The third result is concerned with the relation between the volume growth and the curvature decay. We prove that the curvature decay of a complete noncompact K\"{a}hler manifold with nonnegative curvature operator and with the maximal volume growth is precisely quadratic in certain average sense. | Volume Growth and Curvature Decay of Positively Curved Kähler
manifolds | 14,666 |
The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts. The first consists of isolated examples: the Riemannian symmetric spaces. The second consists of geometries that can occur in continuous families: these include the Calabi-Yau structures and Joyce manifolds of string theory. One may ask how one can weaken the definitions and still obtain similar classifications. We present two closely related suggestions. The classifications for these give isolated examples that are isotropy irreducible spaces, and known families that are the nearly K\"ahler manifolds in dimension 6 and Gray's weak holonomy G$_2$ structures in dimension 7. | Einstein Metrics via Intrinsic or Parallel Torsion | 14,667 |
In this paper, we study a certain cohomology attached to a smooth function, which arose naturally in Poisson geometry. We explain how this cohomology depends on the function, and we prove that it satisfies both the excision and the Mayer-Vietoris axioms. For a regular function we show that the cohomology is related to the de Rham cohomology. Finally, we use it to give a new proof of a well-known result of A. Dimca in complex analytic geometry. | A cohomology attached to a function | 14,668 |
Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper we study a generalization, which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a family of compatible, local, Lagrangian functions. We give several examples, and find the cohomological obstractions to globalization. Then, we extend the connections used in Finsler and Lagrange geometry, while giving an index free presentation of these connections. | Lagrange Geometries on Tangent Manifolds | 14,669 |
We prove that for certain endomorphisms of a nilmanifold N the set S of those points such that the closure of its (forward) orbit contains no periodic points is large in the sense that for any non-empty open set U, the set U\cap S is of full Hausdorff dimension. When the manifold N is a torus, this result is due to S.G.Dani. | Orbits of certain endomorphisms of nilmanifolds and Hausdorff dimension | 14,670 |
In this article, we give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of Lie groups defined by T.Robart [13], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose-Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements. | Structure groups and holonomy in infinite dimensions | 14,671 |
We give a simple interpretation of the adapted complex structure of Lempert-Szoke and Guillemin-Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of $X$ in $TTX$, where $X$ is a real-analytic manifold equipped with a linear connection. We show that the Nahm equations arise naturally in this context: for a connection with zero curvature and arbitrary torsion, the real sections of the twistor space can be obtained by solving Nahm's equations in the Lie algebra of certain vector fields. Finally, we show that, if we start with a metric connection, then our construction yields an SO(3)-invariant hyperk\"ahler metric. | Complexification and hypercomplexification of manifolds with a linear
connection | 14,672 |
We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior and a term that comes from the boundary. This answers the index problem formulated by Mazzeo and Melrose. We give a more precise answer in the case where the base of the boundary fiber bundle is the circle. In particular, for Dirac operators associated to a "product fibered cusp metric", the index is given by the integral of the Atiyah-Singer form in the interior minus the adiabatic limit of the eta invariant of the restriction of the operator to the boundary. | An index formula on manifolds with fibered cusp ends | 14,673 |
The paper investigates higher dimensional analogues of Burago's inequality bounding the area of a closed surface by its total curvature. We obtain sufficient conditions for hypersurfaces in 4-space that involve the Ricci curvature. We get semi-local variants of the inequality holding in any dimension that involve domains with non-vanishing Gauss-Kronecker curvature. The paper also contains inequalities of isoperimetric type involving the total curvature, as well as a "reverse" isoperimetric inequality for spaces with constant curvature. | Volume et courbure totale pour les hypersurfaces de l'espace euclidien | 14,674 |
In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition, see section 2) generalizing the conformal Laplacian, and their associated conformal invariants have been introduced. The conformally covariant powers of the Laplacian form a family $P_{2k}$ with $k \in \mathbb N$ and $k \leq \frac{n}{2}$ if the dimension $n$ is even. Each $P_{2k}$ has leading order term $(- \Delta)^k$ and is equal to $ (- \Delta) ^k$ if the metric is flat. | Non-linear partial differential equations in conformal geometry | 14,675 |
Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants. Those invariants have enabled us to prove a number of striking results for low dimensional manifolds, particularly, 4-manifolds. The theory of Gromov-Witten invariants was established by using solutions of the Cauchy-Riemann equation. These solutions are often refered as pseudo-holomorphic maps which are special minimal surfaces studied long in geometry. It is certainly not the end of applications of nonlinear partial differential equations to geometry. In this talk, we will discuss some recent progress on nonlinear partial differential equations in geometry. We will be selective, partly because of my own interest and partly because of recent applications of nonlinear equations. There are also talks in this ICM to cover some other topics of geometric analysis by R. Bartnik, B. Andrew, P. Li and X.X. Chen, etc. | Geometry and nonlinear analysis | 14,676 |
We study almost K\"ahler manifolds whose curvature tensor satisfies the second curvature condition of Gray (shortly ${\cal{AK}}_2$). This condition is interpreted in terms of the first canonical Hermitian connection. It turns out that this condition forces the torsion of this connection to be parallel in directions orthogonal to the K\"ahler nullity of the almost complex structure. We prove a local structure result for ${\cal{AK}}_2$ manifolds, showing that the basic pieces are manifolds with parallel torsion and special almost K\"ahler manifolds, a class generalizing, to some algebraic extent, the class of 4-dimensional ${\cal{AK}}_2$-manifolds. In the case of parallel torsion, the Einstein condition and the reducibility of the canonical Hermitian connection is studied. | Torsion in almost Kaehler geometry | 14,677 |
We produce a new general family of flat tori in R^4, the first one since Bianchi's classical works in the 19th century. To construct these flat tori, obtained via small perturbation of certain Hopf tori in S^3, we first present a global description of all isometric immersions of R^2 into R^4 with flat normal bundle. | Isometric immersions of R^2 into R^4 and pertubation of Hopf tori | 14,678 |
In this note we specialize and illustrate the ideas developed in the paper math.DG/0201112 of the first author ("Index theory, eta forms, and Deligne cohomology ") in the case of the determinant line bundle. We discuss the surgery formula in the adiabatic limit using the adiabatic decomposition formula of the zeta regularized determinant of the Dirac Laplacian obtained by the second author and K. Wojciechowski. | Determinant bundles, boundaries, and surgery | 14,679 |
CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the ``conformally invariant powers of the Laplacian'' via the Fefferman metric; the powers which arise for these operators are bounded in terms of the dimension. A second family is derived from a CR tractor calculus which is developed here; this family includes operators for every positive power of the sub-Laplacian. This result together with work of Cap, Slovak and Soucek imply in three dimensions the existence of a curved analogue of each such operator in flat space. | CR Invariant powers of the sub-Laplacian | 14,680 |
In this paper, we continue studying the 6-dimensional pseudo-Riemannian space V^6(g_{ij}) with signature [++--], which admits projective motions, i. e. continuous transformation groups preserving geodesics. In particular, we determine a necessary and sufficient condition that the 6-dimensional rigid h-spaces have constant curvature. | Rigid 6-dimensional h-spaces of constant curvature | 14,681 |
The main goal of this paper is to show a counterexample to the following conjecture: {\bf Conjecture} [Meeks, Sullivan]: If $f:M\to \mathbb{R}^3$ is a complete proper minimal immersion where $M$ is a Riemannian surface without boundary and with finite genus, then $M$ is parabolic. We have proved: {\bf Theorem:} There exists $\chi: D\longrightarrow \mathbb{R}^3$, a conformal proper minimal immersion defined on the unit disk. | On the existence of a proper minimal surface in $R^3$ with the conformal
type of a disk | 14,682 |
It is classically known that closed geodesics on a compact Riemann surface with a metric of negative curvature strictly minimize length in their free homotopy class. We'd like to generalize this to Lagrangian submanifolds in K\"ahler manifolds of negative Ricci curvature. The only known result in this direction is a theorem on Y.I. Lee for certain Lagrangian submanifolds in a product of two Riemann surfaces of constant negative curvature. We develop an approach to study this problem in higher dimensions. Along the way we prove some weak results (volume-minimization outside of a divisor) and give a counterexample to global volume-minimization for an immersed minimal Lagrangian submanifold. | Volume minimization for Lagrangian submanifolds in complex manifolds
with negative first Chern class | 14,683 |
We consider some infinitesmal and global deformations of G_2 structures on 7-manifolds. We discover a canonical way to deform a G_2 structure by a vector field in which the associated metric gets "twisted" in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field whose solution would yield a manifold with holonomy G_2. Similarly we consider analogous constructions for Spin(7) structures on 8-manifolds. Some of the results carry over directly, while others do not because of the increased non-linearity of the Spin(7) case. | Deformations of G_2 and Spin(7) Structures on Manifolds | 14,684 |
The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of $\r^3$, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable collection of circles in $\r^3$ passing through one point. It turns out that to be rectifiable all circles need to pass through some other common point. The second main result is a complete description of geometries in $\r^3$ in which all the geodesics are circles. This is a consequence of an extension of Beltrami's theorem by replacing straight lines with circles.} | On Rectification of Circles and an Extension of Beltrami's Theorem | 14,685 |
Let $ l : \Sigma \to X$ be a weakly Lagrangian map of a compact orientable surface $ \Sigma$ in a K\"ahler surface $ X$ which is area minimizing in its homotopy class of maps in $ W^{1,2}(\Sigma, X)$, the Sobolev space of maps of square integrable first derivative. Schoen and Wolfson showed such $ l$ is Lipschitz, and it is smooth except at most at finitely many points of Maslov index 1 or -1. In this note, we observe if in addition c_1(X)[l]=0, $ l$ is smooth everywhere. Here $ c_1(X)$ is the first Chern class of $ X$. | A remark on a theorem of Schoen and Wolfson | 14,686 |
We shall investigate flat surfaces in hyperbolic 3-space with admissible singularities, called `flat fronts'. An Osserman-type inequality for complete flat fronts is shown. When equality holds in this inequality, we show that all the ends are embedded. Moreover, we shall give new examples for which equality holds. | Flat fronts in hyperbolic 3-space | 14,687 |
We study special almost Kaehler manifolds whose curvature tensor satisfies the second curvature condition of Gray. It is shown that for such manifolds, the torsion of the first canonical Hermitian is parallel. This enables us to show that every AK_2-manifold has parallel torsion. Some applications of this result, concerning the existence of orthogonal almost Kaehler structures on spaces of constant curvature are given. | The structure of AK_2-manifolds | 14,688 |
In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold is Lagrangian and almost calibrated by Re\Omega in a Calabi-Yau n-fold (M,\Omega), and T>0 is the first blow-up time of the mean curvature flow, then the tangent cone of the mean curvature flow at a singular point (X,T) is a stationary Lagrangian integer multiplicity current in R\sup 2n with volume density greater than one at X. When n=2, the tangent cone consists of a finite union of more than one 2-planes in R\sup 4 which are complex in a complex structure on R\sup 4. | Singularity of Mean Curvature Flow of Lagrangian Submanifolds | 14,689 |
We present an approach to solvable pseudo-Riemannian symmetric spaces based on papers of M.Cahen, M.Parker and N.Wallach. Thereby we reproduce the classification of solvable symmetric triples of Lorentzian signature $(1,n-1)$ and complete the case of signature $(2,n-2)$. Moreover we discuss the topology of non-simply-connected symmetric spaces. | Solvable Pseudo-Riemannian Symmetric Spaces | 14,690 |
We study the geometric structure of Lorentzian spin manifolds, which admit imaginary Killing spinors. The discussion is based on the cone construction and a normal form classification of skew-adjoint operators in signature $(2,n-2)$. Derived geometries include Brinkmann spaces, Lorentzian Einstein-Sasaki spaces and certain warped product structures. Exceptional cases with decomposable holonomy of the cone are possible. | Imaginary Killing Spinors in Lorenztian Geometry | 14,691 |
We determine the complete conjugate locus along all geodesics parallel or perpendicular to the center (Theorem 2.3). When the center is 1-dimensional we obtain formulas in all cases (Theorem 2.5), and when a certain operator is also diagonalizable these formulas become completely explicit (Corollary 2.7). These yield some new information about the smoothness of the pseudoriemannian conjugate locus. We also obtain the multiplicities of all conjugate points. | Conjugate loci of pseudoriemannian 2-step nilpotent Lie groups with
nondegenerate center | 14,692 |
Pseudo-Riemannian manifolds of balanced signature which are both spacelike and timelike Jordan Osserman nilpotent of order 2 and of order 3 have been constructed previously. In this short note, we shall construct pseudo-Riemannian manifolds of signature (2s,s) for any s (which is at least 2) which are spacelike Jordan Osserman nilpotent of order 3 but which are not timelike Jordan Osserman. Our example and techniques are quite different from known previously both in that they are not in neutral signature and that the manifolds constructed will be spacelike but not timelike Jordan Osserman. | Nilpotent Spacelike Jorden Osserman pseudo-Riemannian manifolds | 14,693 |
Let M be a riemannian manifold. The existence of a spin structure on M, enables to study the topology of M. The obstruction to the existence of the spin structure is given by the second Stiefel-Whitney class. This class is the classifying cocycle of a gerbe. One may expect that the study of this gerbe may have topological applications, for example, one may try to generalize the spinors Lichnerowicz theorem in this setting. On this purpose, we must first prove an Atiyah-Singer theorem for gerbes which is the main goal of this paper. | An Atiyah-Singer theorem for gerbes | 14,694 |
We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT) approach. We show both for bundles and gerbes with connection that there is a one-to-one correspondence between their local description in terms of locally-defined functions and forms and their non-local description in terms of a suitable class of embedded TQFT's. | TQFT's and gerbes | 14,695 |
We announce a new proof of the uniform estimate on the curvature of solutions to the Ricci flow on a compact K\"ahler manifold $M^n$ with positive bisectional curvature. In contrast to the recent work of X. Chen and G. Tian, our proof of the uniform estimate does not rely on the exsitence of K\"ahler-Einstein metrics on $M^n$, but instead on the first author's Harnack inequality for the K\"ahler-Ricc flow, and a very recent local injectivity radius estimate of Perelman for the Ricci flow. | Ricci flow on compact Kähler manifolds of positive bisectional
curvature | 14,696 |
The use of bundle gerbes and bundle gerbe modules is considered as a replacement for the usual theory of Clifford modules on manifolds that fail to be spin. It is shown that both sides of the Atiyah-Singer index formula for coupled Dirac operators can be given natural interpretations using this language and that the resulting formula is still an identity. | Gerbes, Clifford modules and the index theorem | 14,697 |
We consider dimension reduction for solutions of the K\"ahler-Ricci flow with nonegative bisectional curvature. When the complex dimension $n=2$, we prove an optimal dimension reduction theorem for complete translating K\"ahler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reduction theorem for complete ancient solutions of the K\"ahler-Ricci flow with nonnegative bisectional curvature on noncompact complex manifolds under a finiteness assumption on the Chern number $c^n_1$. | On dimension reduction in the Kähler-Ricci flow | 14,698 |
Consider oriented surfaces immersed in $\mathbb R^3.$ Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature $\mathcal K$, given by the product of the principal curvatures $k_1, k_2$ is positive. The leaves of the foliations are the lines of harmonic mean curvature, also called characteristic or diagonal lines, along which the normal curvature of the immersion is given by ${\mathcal K}/{\mathcal H}$, where $ {\mathcal H}=({k_1}+k_2)/2$ is the arithmetic mean curvature. That is, ${\mathcal K}/{\mathcal H}=((1/{k_1} + 1/{k_2})/2)^{-1}$ is the harmonic mean of the principal curvatures $k_1, k_2$ of the immersion. The singularities of the foliations are the umbilic points and parabolic curves, where $k_1 = k_2$ and ${\mathcal K} = 0$, respectively. Here are determined the structurally stable patterns of harmonic mean curvature lines near the umbilic points, parabolic curves and harmonic mean curvature cycles, the periodic leaves of the foliations. The genericity of these patterns is established. This provides the three essential local ingredients to establish sufficient conditions, likely to be also necessary, for Harmonic Mean Curvature Structural Stability of immersed surfaces. This study, outlined towards the end of the paper, is a natural analog and complement for that carried out previously by the authors for the Arithmetic Mean Curvature and the Asymptotic Structural Stability of immersed surfaces. | Harmonic Mean Curvature Lines on Surfaces Immersed in R3 | 14,699 |
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