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A discussion of torsion of Riemannian G-structures leads to a survey of contributions of Alfred Gray and others on almost Hermitian manifolds, G_2-manifolds, curvature identities, volume expansions, plotting geodesics, and the geometry of nilmanifolds. The paper concludes with a new example of a compact 8-manifold with a quaternionic 4-form that is closed but not parallel. | Almost Parallel Structures | 14,500 |
We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures. | Connections and curvings on lifting bundle gerbes | 14,501 |
This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. <p> The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. <p> The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in complex projective n-space. <p> The third remark is that there is a description of the Finsler metrics of constant curvature on the 2-sphere in terms of a Riemannian metric and 1-form on the space of its geodesics. In particular, this allows one to use any (Riemannian) Zoll metric of positive Gauss curvature on the 2-sphere to construct a global Finsler metric of constant positive curvature on the 2-sphere. <p> The fourth remark concerns the generality of the space of (local) Finsler metrics of constant positive flag curvature in dimension n+1>2 . It is shown that such metrics depend on n(n+1) arbitrary functions of n+1 variables and that such metrics naturally correspond to certain torsion-free S^1 x GL(n,R)-structures on 2n-manifolds. As a by-product, it is found that these groups do occur as the holonomy of torsion-free affine connections in dimension 2n, a hitherto unsuspected phenomenon. | Some remarks on Finsler manifolds with constant flag curvature | 14,502 |
We present a classification of SU(2) instantons on $T^2\times\mathbb{R}^2$ according to their asymptotic behaviour. We then study the existence of such instantons for different values of the asymptotic parameters, describing explicitly the moduli space for unit charge. | Classification and existence of doubly-periodic instantons | 14,503 |
In this paper, we study the limiting properties of the $K$ energy for smooth hypersurfaces in the projective spaces. Our result generalizes the result of Ding-Tian (W. Ding and G. Tian. K\"ahler-Einstein metrics and the generalized Futaki invariant. {\em Invent Math}, 110:315-335, 1992.) in the case of hypersurfaces. In particular, we allow the center fiber of a special degeneration (a degeneration by a one-parameter group) to have multiplicity great than 1. | K Energy and K stability on Hypersurfaces | 14,504 |
Let (M,g) be a compact Riemannian manifold with dimension n > 2. The Yamabe problem is to find a metric with constant scalar curvature in the conformal class of g, by minimizing the total scalar curvature. The proof was completed in 1984. Suppose (M',g') and (M'',g'') are compact Riemannian n-manifolds with constant scalar curvature. We form the connected sum M' # M'' of M' and M'' by removing small balls from M' and M'' and joining the S^{n-1} boundaries together. In this paper we use analysis to construct metrics with constant scalar curvature on M' # M''. Our description is quite explicit, in contrast to the general Yamabe case when one knows little about what the metric looks like. There are 9 cases, depending on the signs of the scalar curvature on M' and M'' (positive, negative, or zero). We show that the constant scalar curvature metrics either develop small "necks" separating M' and M'', or one of M', M'' is crushed small by the conformal factor. When both have positive scalar curvature, we construct three different metrics with scalar curvature 1 in the same conformal class. | Constant Scalar Curvature Metrics on Connected Sums | 14,505 |
We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot-Caratheodory metric. We generate a large library of examples of minimal surfaces and use these to show that the solution to the Dirichlet problem need not be unique. Moreover, we show that the minimal surfaces we construct are in fact X-minimal surfaces in the sense of Garofalo and Nhieu. | Minimal surfaces in the Heisenberg group | 14,506 |
A classical result of Sampson and Schoen-Yau in 1978 states that every diffeomorphism between compact hyperbolic Riemann surfaces is homotopic to an harmonic diffeomorphism. As conjectured by Schoen in 1993 and partially proved by Wan in 1992 and Tam-Wan in 1995, we prove in this article that this theorem generalizes to the non compact case : every quasi symmetric homeomorphism of the circle extends to a quasi isometric harmonic diffeomorphism of the hyperbolic plane. This enables to parametrize the universal Teichmuller space by bounded holomorphic quadratric differentials of the hyperbolic plane. | Diffeomorphismes harmoniques du plan hyperbolique | 14,507 |
The spaces of Riemannian metrics on a closed manifold $M$ are studied. On the space ${\mathcal M}$ of all Riemannian metrics on $M$ the various weak Riemannian structures are defined and the corresponding connections are studied. The space ${\mathcal AM}$ of associated metrics on a symplectic manifold $M,\omega$ is considered in more detail. A natural parametrization of the space ${\mathcal AM}$ is defined. It is shown, that ${\mathcal AM}$ is a complex manifold. A curvature of the space ${\mathcal AM}$ and quotient space ${\mathcal AM}/{\mathcal D}_{\omega}$ is found. The finite dimensionality of the space of associated metrics of a constant scalar curvature with Hermitian Ricci tensor is shown. | The space of associated metrics on a symplectic manifold | 14,508 |
This paper, the second of a series, deals with the function space of all smooth K\"ahler metrics in any given closed complex manifold $M$ in a fixed cohomology class. The previous result of the second author \cite{chen991} showed that the space is a path length space and it is geodesically convex in the sense that any two points are joined by a unique path, which is always length minimizing and of class C^{1,1}. This already confirms one of Donaldson's conjecture completely and verifies another one partially. In the present paper, we show first of all, that the space is, as expected, a path length space of non-positive curvature in the sense of A. D. Alexanderov. The second result is related to the theory of extremal K\"ahler metrics, namely that the gradient flow of the K energy is strictly length decreasing on all paths except those induced by a path of holomorphic automorphisms of $M$. This result, in particular, implies that extremal K\"ahler metric is unique up to holomorphic transformations, provided that Donaldson's conjecture on the regularity of geodesic is true. | The Space of Kähler metrics (II) | 14,509 |
We establish a correspondence between Darboux's special isothermic surfaces of type (A,0,C,D) and the solutions of the second order PDE : u\Delta(u)-|\nabla(u)|^{2}+\Phi^{4}=s, s \in R. We then use the classical Darboux transformation for isothermic surfaces to construct a B\"acklund transformation for this equation and prove a superposition formula for its solutions. As an application we discuss 1 and 2-soliton solutions and the corresponding surfaces. | Special Isothermic Surfaces and Solitons | 14,510 |
In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the K\"ahler Ricci flow. The one of main ideas is to find a set of new functionals on curvature tensors such that the Ricci flow is the gradient like flow of these functionals. We successfully find such functionals in case of Kaehler manifolds. On K\"ahler-Einstein manifold with positive scalar curvature, if the initial metric has positive bisectional curvature, we prove that these functionals have a uniform lower bound, via the effective use of Tian's inequality. Consequently, we prove the following theorem: Let $M$ be a K\"ahler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the K\"ahler Ricci flow will converge exponentially fast to a K\"ahler-Einstein metric with constant bisectional curvature. Such a result holds for K\"ahler-Einstein orbifolds. | Ricci flow on Kähler-Einstein manifolds | 14,511 |
The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2) tensor under projective coordinate transformations, so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold M, the section u can be regarded as a metric potential analogous to the local potential in Kahler geometry. If M is compact and u is a negative section of the dual of the tautological bundle whose Hessian is positive definite, then M is projectively equivalent to a quotient of a bounded convex domain in R^n. The same is true if M has a boundary on which u=0. This theorem is analogous to a result of Schoen and Yau in locally conformally flat geometry. The proof uses affine differential geometry techniques developed by Cheng and Yau. | Riemannian Metrics on Locally Projectively Flat Manifolds | 14,512 |
Let $M$ be a manifold and $T^*M$ be the cotangent bundle. We introduce a 1-cocycle on the group of diffeomorphisms of $M$ with values in the space of linear differential operators acting on $C^{\infty} (T^*M).$ When $M$ is the $n$-dimensional sphere, $S^n$, we use this 1-cocycle to compute the first-cohomology group of the group of diffeomorphisms of $S^n$, with coefficients in the space of linear differential operators acting on contravariant tensor fields. | Cohomology of groups of diffeomorphims related to the modules of
differential operators on a smooth manifold | 14,513 |
Recently, for a family of ungraded Dirac operators over some space $B$ J. Lott constructed an index gerbe. In the present paper we show (in analogy to the holonomy formula for the determinant bundle in the graded case) that the holonomy of the index gerbe (a priori a hermitean line bundle with connection over the free loop space $LB$) coincides with an adiabatic limit of determinant bundles of an associated family of graded Dirac operators over $LB$. | Transgression of the index gerbe | 14,514 |
In this paper, we observe that the brane functional studied in hep-th/9910245 can be used to generalize some of the works that Schoen and I [4] did many years ago. The key idea is that if a three dimensional manifold M has a boundary with strongly positive mean curvature, the effect of this mean curvature can influence the internal geometry of M. For example, if the scalar curvature of M is greater than certain constant related to this boundary effect, no incompressible surface of higher genus can exist. | Geometry of three manifolds and existence of Black Hole due to boundary
effect | 14,515 |
It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces. | Leafwise Holomorphic Functions | 14,516 |
We describe a simple way of constructing torus fibrations $T^3\to X\to S^3$ which degenerate canonically over a knot or link in $S^3$. We show that the topological invariants of $X$ can be computed algebraically from the monodromy representation of the fibration. We use this to obtain some new $T^3$-fibrations $S^3\times S^3\to S^3$ and $(S^3\times S^3)#(S^3\times S^3)#(S^4\times S^2) \to S^3$ whose discriminant locus is a torus knot. | $T^3$-fibrations on compact six-manifolds | 14,517 |
We classify, up to a local isometry, all non-Kahler almost Kahler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kahler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure. | Local models and integrability of certain almost Kahler 4-manifolds | 14,518 |
We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then the product of E and R^p admits a complete metric of positive Ricci curvature for all large p. | Metrics of positive Ricci curvature on bundles | 14,519 |
We show the total space of the canonical line bundle $\mathbb{L}$ of a Kahler-Einstein manifold $X^n$ supports integrable $SU(n+1)$ structures, or Calabi-Yau structures. The canonical real line bundle $L \subset \mathbb{L}$ over a minimal Lagrangian submanifold $M \subset X$ is calibrated in this setting and hence can be considered as the special Lagrangian lift of $M$. For the integrable $G_2$ and $Spin(7)$ structures on spin bundles and bundles of anti-self-dual 2-forms on self-dual Einstein 4-manifolds constructed by Bryant and Salamon, minimal surfaces with vanishing complex quartic form (super-minimal) admit lifts which are calibrated, i.e., associative, coassociative or Cayley respectively. The lifts in this case can be considered as the tangential lifts or normal lifts of the minimal surface adapted to the quaternionic bundle structure. | On the Lifts of Minimal Lagrangian Submanifolds | 14,520 |
Let \Sigma be a compact oriented surface immersed in a four dimensional K\"ahler-Einstein manifold M. We consider the evolution of \Sigma in the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel K\"ahler forms \omega' and \omega'' that determine different orientations and\Sigma is symplectic with respect to both \omega' and \omega'', we prove the mean curvature flow of \Sigma exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity. | Mean Curvature Flow of Surfaces in Einstein Four-Manifolds | 14,521 |
Let f:\Sigma_1 --> \Sigma_2 be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of f can be viewed as a Lagrangian submanifold in \Sigma_1\times \Sigma_2. This article discusses a canonical way to deform f along area preserving diffeomorphisms. This deformation process is realized through the mean curvature flow of the graph of f in \Sigma_1\times \Sigma_2. It is proved that the flow exists for all time and the map converges to a canonical map. In particular, this gives a new proof of the classical topological results that O(3) is a deformation retract of the diffeomorphism group of S^2 and the mapping class group of a Riemman surface of positive genus is a deformation retract of the diffeomorphism group . | Deforming Area Preserving Diffeomorphism of Surfaces by Mean Curvature
Flow | 14,522 |
We consider complete nearly K\"ahler manifolds with the canonical Hermitian connection. We prove some metric properties of strict nearly K\"ahler manifolds and give a sufficient condition for the reducibility of the canonical Hermitian connection. A holonomic condition for a nearly K\"ahler manifold to be a twistor space over a quaternionic K\"ahler manifold is given. This enables us to give classification results in 10-dimensions. | On nearly Kaehler geometry | 14,523 |
We study the mechanisms of the non properness of the action of the group of diffeomorphisms on the space of Lorentzian metrics of a compact manifold. In particular, we prove that nonproperness entails the presence of lightlike geodesic foliations of codimension 1. On the 2-torus, we prove that a metric with constant curvature along one of its lightlike foliation is actually flat. This allows us to show that the restriction of the action to the set of non-flat metrics is proper and that on the set of flat metrics of volume 1 the action is ergodic. Finally, we show that, contrarily to the Riemannian case, the space of metrics without isometries is not always open. | Dynamical properties of the space of Lorentzian metrics | 14,524 |
We first prove a general gluing theorem which creates new nondegenerate constant mean curvature surfaces by attaching half Delaunay surfaces with small necksize to arbitrary points of any nondegenerate CMC surface. The proof uses the method of Cauchy data matching from \cite{MP}, cf. also \cite{MPP}. In the second part of this paper, we develop the consequences of this result and (at least partially) characterize the image of the map which associates to each complete, Alexandrov-embedded CMC surface with finite topology its associated conformal structure, which is a compact Riemann surface with a finite number of punctures. In particular, we show that this `forgetful' map is surjective when the genus is zero. This proves in particular that the CMC moduli space has a complicated topological structure. These latter results are closely related to recent work of Kusner \cite{Ku}. | The conformal theory of Alexandrov embedded constant mean curvature
surfaces in $R^3$ | 14,525 |
In the paper [4] is presented a theory which unifies the gravitation theory and the mechanical effects, which is different from the Riemannian theories like GTR. Moreover it is built in the style of the electomagnetic field theory. This paper is a continuation of [4] such that the complex variant of that theory yields to the required unification of gravitation and electromagnetism. While the gravitational field is described by a scalar potential $\mu $, taking a complex value of $\mu $ we obtain the unification theory. For example the electric field appears to be imaginary 3-vector field of acceleration, the magnetic field appears to be imaginary 3-vector of angular velocity and the imaginary part of a complex mass is just electric charge of the particle. | On the Unification of Gravitational and Electromagnetic Fields | 14,526 |
We will introduce two notions of compatibility bettwen pseudo-Riemannian metric and Poisson structure using the notion of contravariant connection introduced by Fernandes R. L., we will study some proprities of manifold endowed with such compatible structures an we will give some examples. | Compatibility betwenn pseudo-riemannian structure and Poisson structure | 14,527 |
This is the writeup of a lecture given at the May Wisconsin workshop on mathematical aspects of orbifold string theory. In the first part of this lecture, we review recent work on discrete torsion, and outline how it is currently understood in terms of the B field. In the second part of this lecture, we discuss the relationship between quotient stacks and string orbifolds. | Discrete Torsion, Quotient Stacks, and String Orbifolds | 14,528 |
In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomegeneity one. We characterize them as the only minimal Lagrangian submanifolds in CH^n foliated by umbilical hypersurfaces of Lagrangian subspaces RH^n of CH^n. Several suitable generalizations of the above construction allow us to get new families of minimal Lagrangian submanifolds in CH^n from curves in CH^1 and (n-1)-dimensional minimal Lagrangian submanifolds of the complex space forms CP^n-1, CH^n-1 and C^n-1. Similar constructions are made in the complex projective space. | Minimal Lagrangian submanifolds in the complex hyperbolic space | 14,529 |
Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of an expression in the curvature and its first two covariant derivatives, of the bundle of 2-forms Whitney sum 1-forms is nonnegative. The idea is to consider the 2-form which minimizes the associated quadratic form to obtain a symmetric 2-tensor. A long but straightforward computation implies this 2-tensor is a subsolution to heat-type equation. A standard application of the maximum principle implies the result. | On an alternate proof of Hamilton's matrix Harnack inequality for the
Ricci flow | 14,530 |
This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of the formal solution of a boundary problem at infinity for the Laplacian in the Poincare metric associated to the conformal structure. This gives an easy proof of the result of Graham-Zworski that the log coefficient in the volume expansion of a Poincare metric is a multiple of the integral of the Q-curvature, and leads to a definition of a non-local version of the Q-curvature in odd dimensions. | Q-Curvature and Poincare Metrics | 14,531 |
Let $(M,g)$ be a pseudo-Riemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1-cocycles on the group of diffeomorphisms of $M$ related to the modules of linear differential operators. As operators, these derivatives do not depend on the rescaling of the metric $g.$ In particular, if the manifold $(M,g)$ is conformally flat, these derivatives vanish on the conformal group $\Og(p+1,q+1),$ where $\mathrm{dim} (M)=p+q.$ This work is a continuation of [1,4] where the Schwarzian derivative was defined on a manifold endowed with a projective connection. | Conformal Schwarzian derivatives and conformally invariant quantization | 14,532 |
This paper goes some way in explaining how to construct an integrable hierarchy of flows on the space of conformally immersed tori in n-space. These flows have first occured in mathematical physics -- the Novikov-Veselov and Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of the Dirac operator. Later, using spinorial representations of surfaces, the same flows were interpreted as deformations of surfaces in 3- and 4-space preserving the Willmore energy. This last property suggest that the correct geometric setting for this theory is Moebius invariant surface geometry. We develop this view point in the first part of the paper where we derive the fundamental invariants -- the Schwarzian derivative, the Hopf differential and a normal connection -- of a conformal immersion into n-space together with their integrability equations. To demonstrate the effectivness of our approach we discuss and prove a variety of old and new results from conformal surface theory. In the the second part of the paper we derive the Novikov-Veselov and Davey-Stewartson flows on conformally immersed tori by Moebius invariant geometric deformations. We point out the analogy to a similar derivation of the KdV hierarchy as flows on Schwarzian's of meromorphic functions. Special surface classes, e.g. Willmore surfaces and isothermic surfaces, are preserved by the flows. | Schwarzian Derivatives and Flows of Surfaces | 14,533 |
We investigate the local geometry of a class of K\"ahler submanifolds $M \subset \R^n$ which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the $(1,1)$-part (i.e. the $dz_id\bar z_j$-components) of the second fundamental form $\alpha$, which we call the pluri-mean curvature. We show that these K\"ahler submanifolds are characterized by the existence of an associated family of isometric submanifolds with rotated second fundamental form. Of particular interest is the isotropic case where this associated family is trivial. We also investigate the properties of the corresponding Gauss map which is pluriharmonic. | Kaehler submanifolds with parallel pluri-mean curvature | 14,534 |
We define a renormalized characteristic class for Einstein asymptotically complex hyperbolic (ACHE) manifolds of dimension 4: for any such manifold, the polynomial in the curvature associated to the characteristic class euler-3signature is shown to converge. This extends a work of Burns and Epstein in the Kahler-Einstein case. This extends a work of Burns and Epstein in the Kahler-Einstein case. We also define a new global invariant for any 3-dimensional pseudoconvex CR manifold, by a renormalization procedure of the eta invariant of a sequence of metrics which approximate the CR structure. Finally, we get a formula relating the renormalized characteristic class to the topological number euler-3signature and the invariant of the CR structure arising at infinity. | A Burns-Epstein invariant for ACHE 4-manifolds | 14,535 |
The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\gencomp (E)$ as the set of Clifford bundles $E'$ which have finite distance to $E$. If $D$, $D'$ are the associated generalized Dirac operators, we prove for the pair $(D,D')$ relative index theorems, define relative $\zeta$-- and $\eta$--functions, relative determinants and in the case of $D=\Delta$ relative analytic torsion. To define relative $\zeta$-- and $\eta$--functions, we assume additionally that the essential spectrum of $D^2$ has a gap above zero. | Relative Zeta Functions, Determinants, Torsion, Index Theorems and
Invariants for Open Manifolds | 14,536 |
Three definitions of a differential form on a tangent structure are considere. It is proved that the (covariant) definition given by Souriau (as a collection of forms indexed by the plaques) is equivalent to a smooth section of the corresponding vector bundle if the space does not have transverse points. | The definition of a differential form on a tangent structure | 14,537 |
This is the first of three papers math.DG/0111326, math.DG/0204343 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using analytic methods. The three papers are surveyed in math.DG/0206016. Let N be such a U(1)-invariant SL 3-fold. Then |z_1|^2-|z_2|^2=2a on N for some real number a. Locally, N can be written as a kind of graph of functions u,v : R^2 --> R satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy. When a is nonzero, u,v are always smooth and N is always nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, which correspond to singular points of N. This paper focusses on the nonsingular case, when a is nonzero. We prove analogues for our nonlinear Cauchy-Riemann equation of well-known results in complex analysis. In particular, we prove existence and uniqueness for solutions of two Dirichlet problems derived from it. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C^3, with two kinds of boundary conditions. In the sequels we extend these results to the singular case a=0. The next paper math.DG/0111326 proves existence and uniqueness of continuous weak solutions to the two Dirichlet problems when a=0. This gives existence and uniqueness of a large class of singular U(1)-invariant SL 3-folds in C^3, with boundary conditions. The final paper math.DG/0204343 studies the nature of the singularities that arise, and constructs U(1)-invariant special Lagrangian fibrations of open sets in C^3. | U(1)-invariant special Lagrangian 3-folds I. Nonsingular solutions | 14,538 |
This is the second of three papers math.DG/0111324, math.DG/0204343 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using analytic methods. The three papers are surveyed in math.DG/0206016. If N is such a 3-fold then |z_1|^2-|z_2|^2=2a on N for some real number a. Locally, N can be written as a kind of graph of functions u,v : R^2 --> R satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy. The first paper math.DG/0111324 studied the case when a is nonzero. Then u,v are smooth and N is nonsingular. It proved existence and uniqueness for solutions of two Dirichlet problems derived from the equations on u,v. This implied existence and uniqueness for a large class of nonsingular U(1)-invariant SL 3-folds in C^3, with boundary conditions. In this paper and its sequel math.DG/0204343 we focus on the case a=0. Then the nonlinear Cauchy-Riemann equation is not always elliptic. Because of this there may be points (x,0) where u,v are not differentiable, corresponding to singular points of N. This paper is concerned largely with technical analytic issues, and the sequel with the geometry of the singularities of N. We prove a priori estimates for derivatives of solutions of the nonlinear Cauchy-Riemann equation, and use them to show existence and uniqueness of weak solutions u,v to the two Dirichlet problems when a=0, which are continuous and weakly differentiable. This gives existence and uniqueness for a large class of singular U(1)-invariant SL 3-folds in C^3, with boundary conditions. | U(1)-invariant special Lagrangian 3-folds II. Existence of singular
solutions | 14,539 |
We prove that every continuous function on a separable infinite-dimensional Hilbert space X can be uniformly approximated by smooth functions with no critical points. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of X can be separated by a one-codimensional smooth manifold which is a level set of a smooth function with no critical points; this fact may be viewed as a nonlinear analogue of the geometrical version of the Hahn-Banach theorem. In particular, every closed set in X can be uniformly approximated by open sets whose boundaries are smooth one-codimensional submanifolds of X. Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space X with any smooth manifold M modelled on X. | Uniform approximation of continuous functions by smooth functions with
no critical points on Hilbert manifolds | 14,540 |
We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps. | Harmonic maps and the topology of conformally compact Einstein manifolds | 14,541 |
This paper constructs a family of conformally invariant differential operators acting on spinor densities with leading part a power of the Dirac operator. The construction applies for all powers in odd dimensions, and only for finitely many powers in even dimensions. These operators arise naturally as obstructions to formal solution of the Dirac equation on the Fefferman-Graham ambient space with prescribed boundary conditions. | Conformally invariant powers of the ambient Dirac operator | 14,542 |
We review the basic definitions and properties concerning smooth structures, convenient spaces, diffeological spaces and tangent structures. The relation betwen them is described. A tangent structure is constructed for each pre-convenient space. This one is proved to be convenient iff the space and the tangent fibres coincide. Some basic comments on the relation with manifolds are given. | Differentiability, Convenient Spaces and Smooth Diffeologies | 14,543 |
One knows that the large time heat decay exponent on a nilpotent group is given by half the growing rate of the volume of its large balls. This work deals with the similar problem of trying to interpret geometrically the heat decay on (one) forms. We will show how it is (partially) related to the depth of the relations required to define the group. The tools used apply in general on Carnot-Caratheodory manifolds. | Around heat decay on forms and relations of nilpotent Lie groups | 14,544 |
We make a study of Poisson structures of T*M which are graded structures when restricted to the fiberwise polynomial algebra, and give examples. A class of more general graded bivector fields which induce a given Poisson structure w on the base manifold M is constructed. In particular, the horizontal lifting of a Poisson structure from M to T*M via connections gives such bivector fields and we discuss the conditions for these lifts to be Poisson bivector fields and their compatibility with the canonical Poisson structure on T*M. Finally, for a 2-form on a Riemannian manifold, we study the conditions for some associated 2-forms on T*M to define Poisson structures on cotangent bundles. | Poisson Structures on Cotangent Bundles | 14,545 |
We prove that every Einstein metric on the unit ball B^4 of C^2, asymptotic to the Bergman metric, is equal to it up to a diffeomorphism. We need a solution of Seiberg--Witten equations in this infinite volume setting. Therefore, and more generally, if M^4 is a manifold with a CR-boundary at infinity, an adapted spinc-structure which has a non zero Kronheimer--Mrowka invariant and an asymptotically complex hyperbolic Einstein metric, we produce a solution of Seiberg--Witten equations with an strong exponential decay property. | Rigidité d'Einstein du plan hyperbolique complexe | 14,546 |
The spectrum of the Laplace-Dolbeault operator for any line bundle with parallel curvature on a flat complex torus is computed. The Ray-Singer analytic torsion is then deduced, generalizing thus Bost's result for ample line bundles and Ray-Singer's ones for flat bundles, of which we a geometric interpretation is given. | Le spectre et la torsion analytique des fibres en droites sur les tores
complexes | 14,547 |
We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from abelian hypercomplex structures. Furthermore, we use a correspondence between abelian hypercomplex structures and subspaces of ${\frak sp}(n)$ to produce continuous families of compact and noncompact of manifolds carrying non isometric HKT structures. Finally, geometrical properties of invariant HKT structures on 2-step nilpotent Lie groups are obtained. | Hyperkähler torsion structures invariant by nilpotent Lie groups | 14,548 |
We shall obtain unobstructed deformations of four geometric structures: Calabi-Yau, HyperK\"ahler, $\G$ and Spin(7) structures in terms of closed differential forms (calibrations). We develop a direct and unified construction of smooth moduli spaces of these four geometric structures and show that the local Torelli type theorem holds in a systematic way. | Moduli spaces of topological calibrations, Calabi-Yau, HyperKähler,
$G_2$ and Spin(7) structures | 14,549 |
Given an smooth function $K <0$ we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genus $g>1$. We do so by minimizing an appropriate functional using elementary analysis. In particular for $K$ a negative constant, this provides an elementary proof of the uniformization theorem for compact Riemann surfaces of genus $g >1$. | A variational proof for the existence of a conformal metric with
preassigned negative Gaussian curvature for compact Riemann surfaces of genus
$>1$ | 14,550 |
We prove the existence of complete, embedded, constant mean curvature 1 surfaces in 3 dimensional hyperbolic space when g, the genus of the surface, and n, the number of ends of the surface, satisfy either g=0 and $n\geq 1$ or $g \geq 1$ and $2n\geq g+5$. The surfaces are all regular points of their corresponding moduli space. | Attaching handles to Bryant surfaces | 14,551 |
In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time--dependent symmetries. In particular we describe how that the finite dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time--dependent Hamiltonian function. We also present a very general Lagrangian formulation of the procedure of reduction. As an application we consider the case of the Painlev\'e equations PI, PII, PIII, PVI and also certain higher order systems appeared in the theory of Frobenius manifolds. | On the Hamiltonian and Lagrangian structures of time-dependent
reductions of evolutionary PDEs | 14,552 |
We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally conformal flat manifolds when $k \neq n/2$. | A fully nonlinear conformal flow on locally conformally flat manifolds | 14,553 |
We consider a connected smooth $n$-dimensional manifold $M$ endowed with a volume form $\Omega$, and we show that an open subset $U$ of $R^n$ of Lebesgue measure $\Vol (U)$ embeds into $M$ by a smooth volume preserving embedding whenever the volume condition $\Vol (U) \le \Vol (M,\Omega)$ is met. | Volume preserving embeddings of open subsets of $R^n$ into manifolds | 14,554 |
The action of origin-preserving diffeomorphisms on a space of jets of symmetric connections is considered. Dimensions of moduli spaces of generic connections are calculated. Poincar\'e series of the geometric structure of symmetric connection is constructed, and shown to be a rational function. | Moduli space of symmetric connections | 14,555 |
We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres $\scriptstyle{\Sigma^{2n+1}}$ the moduli space of Sasakian structures has infinitely many positive components determined by inequivalent underlying contact structures. We also prove the existence of Sasakian metrics with positive Ricci curvature on each of the known $\scriptstyle{2^{2m}}$ distinct diffeomorphism types of homotopy real projective spaces in dimension $4m+1$. | Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature | 14,556 |
In this article I show that every special Lagrangian cone in C^3 determines, and is determined by, a primitive harmonic surface in the 6-symmetric space SU_3/SO_2. For cones over tori, this allows us to use the classification theory of harmonic tori to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some examples found recently by Joyce are put into this context. | Special Lagrangian cones in C^3 and primitive harmonic maps | 14,557 |
The minimal surface equation $Q$ in the second order contact bundle of $R^3$, modulo translations, is provided with a complex structure and a canonical vector-valued holomorphic differential form $Omega$ on $Q\0$. The minimal surfaces $M$ in $R^3$ correspond to the complex analytic curves $C$ in $Q$, where the derivative of the Gauss map sends $M$ to $C$, and $M$ is equal to the real part of the integral of $\Omega$ over $C$. The complete minimal surfaces of finite topological type and with flat points at infinity correspond to the algebraic curves in $Q$. | Second order Contact of Minimal Surfaces | 14,558 |
We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension at least 3 on a closed Riemannian spin manifold, then the Dirac spectrum changes arbitrarily little provided the metric on the manifold after surgery is chosen properly. | Surgery and the Spectrum of the Dirac Operator | 14,559 |
We prove lower Dirac eigenvalue bounds for closed surfaces with a spin structure whose Arf invariant equals 1. Besides the area only one geometric quantity enters in these estimates, the spin-cut-diameter which depends on the choice of spin structure. It can be expressed in terms of various distances on the surfaces or, alternatively, by stable norms of certain cohomology classes. In case of the 2-torus we obtain a positive lower bound for all Riemannian metrics and all nontrivial spin structures. The corresponding estimate also holds for the $L^2$-spectrum of the Dirac operator on a noncompact complete surface of finite area. As a corollary we get positive lower bounds on the Willmore integral for all 2-tori embedded in $R^3$. | Dirac eigenvalue estimates on surfaces | 14,560 |
A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is `parametrised' in some specific way by a vector bundle map from a prescribed vector bundle over M into TM. Some basic properties of these generalised connections are investigated. Special attention is paid to the class of linear connections over a vector bundle map. It is pointed out that not only the more familiar types of connections encountered in the literature, but also the recently studied Lie algebroid connections, can be recovered as special cases within this more general framework. | Generalised connections over a vector bundle map | 14,561 |
Using Weitzenb\"ock techniques on any compact Riemannian spin manifold we derive a general inequality depending on a real parameter and joining the spectrum of the Dirac operator with terms depending on the Ricci tensor and its first covariant derivatives. The discussion of this inequality yields vanishing theorems for the kernel of the Dirac operator $D$ and new lower bounds for the spectrum of $D^2$ if the Ricci tensor satisfies certain conditions. | A relation between the Ricci tensor and the spectrum of the Dirac
operator | 14,562 |
In this paper, which is a sequel to math.DG/9902111, we analyze the limit of the p-form Laplacian under a collapse with bounded sectional curvature and bounded diameter to a singular limit space. As applications, we give results about upper and lower bounds on the j-th eigenvalue of the p-form Laplacian, in terms of sectional curvature and diameter. | Collapsing and the Differential Form Laplacian : The Case of a Singular
Limit Space | 14,563 |
We introduce self-dual manifolds and show that they can be used to encode mirror symmetry for affine-K\"{a}hler manifolds and for elliptic curves. Their geometric properties, especially the link with special lagrangian fibrations and the existence of a transformation similar to the Fourier-Mukai functor, suggest that this approach may be able to explain mirror symmetry also in other situations | Mirror symmetry and self-dual manifolds | 14,564 |
The space ${\mathcal A}$ of almost complex structures on a closed manifold $M$ is studied. A natural parametrization of the space ${\mathcal A}$ is defined. It is shown, that ${\mathcal A}$ is a infinite dimensional complex weak Pseudo-Riemannian manifold. A curvature of the space ${\mathcal A}$ is found. The space ${\mathcal A}_\omega$ of associated almost complex structures on a symplectic manifold $M,\omega$ and space ${\mathcal AO}$ of orthogonal almost complex structures on a Riemannian manifold $M,g_0$ are considered in more detail. | On the space of almost complex structures | 14,565 |
In this thesis we consider a way to construct a rich family of compact Riemann Surfaces in a combinatorial way. Given a 3-regualr graph with orientation, we construct a finite-area hyperbolic Riemann surface by gluing triangles according to the combinatorics of the graph. We then compactify this surface by adding finitely many points. We discuss this construction by considering a number of examples. In particular, we see that the surface depends in a strong way on the orientation. We then consider the effect the process of compactification has on the hyperbolic metric of the surface. To that end, we ask when we can change the metric in the horocycle neighbourhoods of the cusps to get a hyperbolic metric on the compactification. In general, the process of compactification can have drastic effects on the hyperbolic structure. For instance, if we compactify the 3-punctured sphere we lose its hyperbolic structure. We show that when the cusps have lengths > 2\pi, we can fill in the horocycle neighbourhoods and retain negative curvature. Furthermore, the last condition is sharp. We show by examples that there exist curves arbitrarily close to horocycles of length 2\pi, which cannot be so filled in. Such curves can even be taken to be convex. | Riemann Surfaces and 3-Regular Graphs | 14,566 |
We consider the problem of extending a conformal metric of negative curvature, given outside a neighbourhood of 0 in the unit disk $\DD$, to a conformal metric of negative curvature in $\DD$. We give conditions under which such an extension is possible, and also give obstructions to such an extension. The methods we use are based on a maximum principle and the Ahlfors--Schwarz Lemma. We apply these considerations to compactification of Riemann surfaces. | Conformal Extentsion of Metrics of Negative Curvature | 14,567 |
A transitive smooth action of a connected Lie group G on a manifold M is called almost primitive (resp. primitive) if G doesn't contain any proper subgroup (resp. any proper normal subgroup) whose induced action on M is transitive as well. The aim of the present work is to investigate some combinatory properties of symplectic actions of completely solvable Lie groups. | Sur Les Actions Symplectiques Quasi-Primitives | 14,568 |
The present work is devoted to compact completely solvable solvmanifolds which admit Kahlerian metrics whose Kahler forms are homogeneous. In particular, we show that such manifolds are diffeomorphic to flat tori. Our proof is based on Dynkin diagrams associated to left invariant closed 2-forms in completely solvable Lie groups. These diagrams are shown to be closely related to Lagrangian foliations,primitive degrees of homogeneous spaces, and geometric structures such as $(\bkR^ n, Aff (\bkR^ n))$-structures on symplectic homogeneous spaces. | Sur la Conjecture Des Tores Plats | 14,569 |
This work is devoted to an intrinsic cohomology theory of Koszul-Vinberg algebras and their modules. Our results may be regarded as improvements of the attempt by Albert Nijenhuis in [NA]. The relationships between the cohomology theory developed here and some classical problems are pointed out, e.g. extensions of algebras and modules, and deformation theory. The real Koszul-Vinberg cohomology of locally flat manifolds is initiated. Thus regarding the idea raised by M. Gerstenhaber we can state : The category of KV-algebras has its proper cohomology theory. | The cohomology of Koszul-Vinberg algebras | 14,570 |
We analyse the relationship between the components of the intrinsic torsion of an SU(3) structure on a 6-manifold and a G_2 structure on a 7-manifold. Various examples illustrate the type of SU(3) structure that can arise as a reduction of a metric with holonomy G_2. | The intrinsic torsion of SU(3) and G_2 structures | 14,571 |
We consider strict and complete nearly Kaehler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. We show that a strict and complete nearly Kaehler is locally a Riemannian product of homogenous nearly Kaehler spaces, twistor spaces over quaternionic Kaehler manifolds and 6-dimensional nearly Kaehler manifolds. As an application we obtain structure results for totally geodesic Riemannian foliations admitting a compatible Kaehler structure. Finally, we obtain a classification result for the homogenous case, reducing a conjecture of Wolf and Gray to its 6-dimensional form. | Nearly Kaehler geometry and Riemannian foliations | 14,572 |
We study the Lie algebra of infinitesimal isometries on compact Sasakian and K--contact manifolds. On a Sasakian manifold which is not a space form or 3--Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K--contact structure, we prove that there exists a Killing vector field of constant length which is not an infinitesimal automorphism of the structure if and only if the manifold is obtained from the Konishi bundle of a compact pseudo--Riemannian quaternion--Kaehler manifold after changing the sign of the metric on a maximal negative distribution. We also prove that non--regular Sasakian manifolds are not homogeneous and construct examples with cohomogeneity one. Using these results we obtain in the last section the classification of all homogeneous Sasakian manifolds. | Symmetries of Contact Metric Manifolds | 14,573 |
We deal with smooth real manifolds as well as complex analytic manifolds as well. It is well known that the concept of star product is powerful enough to produce all Poisson structures on real manifolds. According to [BdM] it is not known whether holomorphic star products exist on complex analytic manifolds. The main purpose of this paper is to show that the concept of homology of Koszul-Vinberg algebroids on smooth (resp. complex analytic) manifolds is an effective tool to produce smooth (resp complex analytic) Poisson structures on smooth (resp. complex analytic) manifolds. We also study some invariants of contact structures which arise from the associated Koszul-Vinberg algebroids. | The homology theory of Koszul-Vinberg algebroids and Poisson manifolds
II | 14,574 |
The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on constructions for compact 7- and 8-manifolds with holonomy G2 and Spin(7). The simplest such constructions work by using techniques from complex geometry and Calabi-Yau analysis to resolve the singularities of a torus orbifold T^7/G or T^8/G, for G a finite group preserving a flat G2 or Spin(7)-structure on T^7 or T^8. There are also more complicated constructions which begin with a Calabi-Yau manifold or orbifold. All the material in this paper is covered in much more detail in the author's book, "Compact manifolds with special holonomy", Oxford University Press, 2000. | Constructing compact manifolds with exceptional holonomy | 14,575 |
Several classes of irreducible orthogonal representations of compact Lie groups that are of importance in Differential Geometry have the property that the second osculating spaces of all of their nontrivial orbits coincide with the representation space. We say that representations with this property are of class O^2. Our approach in the present paper will be to find restrictions on the class O^2 and then apply them to classify variationally complete and taut representations. The known classifications of cohomogeneity one and two orthogonal representations and more generally of polar representations will also follow easily. | Representations of compact Lie groups and the osculating spaces of their
orbits | 14,576 |
Let G be a Lie group with Lie algebra $ \Cal G: = T_\epsilon G$ and $T^*G = \Cal G^* \rtimes G$ its cotangent bundle considered as a Lie group, where G acts on $\Cal G^*$ via the coadjoint action. We show that there is a 1-1 correspondance between the skew-symmetric solutions $r\in \wedge^2 \Cal G$ of the Classical Yang-Baxter Equation in G, and the set of connected Lie subgroups of $T^*G$ which carry a left invariant affine structure and whose Lie algebras are lagrangian graphs in $ \Cal G \oplus \Cal G^*$. An invertible solution r endows G with a left invariant symplectic structure and hence a left invariant affine structure. In this case we prove that the Poisson Lie tensor $\pi := r^+ - r^-$ is polynomial of degree at most 2 and the double Lie groups of $(G,\pi)$ also carry a canonical left invariant affine structure. In the general case of (non necessarly invertible) solutions r, we supply a necessary and suffisant condition to the geodesic completness of the associated affine structure | Classical Yang-Baxter Equation and Left Invariant Affine Geometry on Lie
Groups | 14,577 |
An explicit seminorm $||f||_{#}$ on the vector space of Chow vectors of projective varieties is introduced, and shown to be a generalized Mabuchi energy functional for Chow varieties. The singularities of the Chow varieties give rise to currents supported on their singular loci, while the regular parts are shown to reproduce the Mabuchi energy functional of the corresponding projective variety. Thus the boundedness from below of the Mabuchi functional, and hence the existence of K\"ahler-Einstein metrics, is related to the behavior of the current $[Y_s]$ and the seminorm $||f||_{#}$ along the orbits of $SL(N+1,{\bf C})$. | Stability, energy functionals, and Kähler-Einstein metrics | 14,578 |
For a Riemannian closed spin manifold and under some topological assumption (non-zero $\hat{A}$-genus or enlargeability in the sense of Gromov-Lawson), we give an optimal upper bound for the infimum of the scalar curvature in terms of the first eigenvalue of the Laplacian. The main difficulty lies in the study of the odd-dimensional case. On the other hand, we study the equality case for the closed spin Riemannian manifolds with non-zero $\hat{A}$-genus. This work improves an inequality which was first proved by K. Ono in 1988. | An optimal inequality between scalar curvature and spectrum of the
Laplacian | 14,579 |
The requirement that a (non-Einstein) K\"ahler metric in any given complex dimension $m>2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m>2$, on three real parameters along with an arbitrary K\"ahler-Einstein metric $h$ in complex dimension $m-1$. We provide an explicit description of all these local-isometry types, for any given $h$. That result is derived from a more general local classification theorem for metrics admitting functions we call {\it special K\"ahler-Ricci potentials}. | Local classification of conformally-Einstein Kähler metrics in higher
dimensions | 14,580 |
The Oscillator Groups,$\G_\lambda,$ are the only solvable, non commutative, simply connected Lie groups to admit a Lorentzian bi-invariant metric. For these groups, we give sufficient conditions for a left-invariant pseudo-Riemannian metric to be complete, we determine the group of isometries, we exhibit a left-invariant affine structure and prove that it is not Lorentzian. As an application, we provide new examples of compact pseudo-Riemannian (sometimes Lorentzian) locally symmetric, occasionally affine, manifolds, complete or incomplete. | Géométrie des groupes oscillateurs et variétés localement
symétriques | 14,581 |
We study Riemannian foliations with complex leaves on Kaehler manifolds. The tensor T, the obstruction to the foliation be totally geodesic, is interpreted as a holomorphic section of a certain vector bundle. This enables us to give classification results when the manifold is compact. | Rigidity of Riemannian foliations with complex leaves on Kaehler
manifolds | 14,582 |
We generalize the spinorial characterization of isometric immersions of surfaces in R^3 given by T. Friedrich (On the spinor representation of surfaces in Euclidean 3-space, J. Geom. Phys. 28 (1998)) to surfaces in S^3 and H^3. The main argument is the interpretation of the energy-momentum tensor associated with a special spinor field as a second fundamental form. It turns out that such a characterization of isometric immersions in terms of a special section of the spinor bundle also holds in the case of hypersurfaces in the Euclidean 4-space. | Surfaces in S^3 and H^3 via Spinors | 14,583 |
We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $\delta$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values ${\eta}(\delta_t,0)$ and $t{\zeta}(\delta_t,0)$ are smooth functions of $t$ at $t=0$, and we identify their values at $t=0$ with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions ${\eta}(\delta_t,s)$ and $t{\zeta}(\delta_t,s)$ are shown to extend smoothly to $t=0$ for all values of $s$. After normalizing with a Gamma factor, the zeta function satisfies in the adiabatic limit an identity reminiscent of the Riemann zeta function, while the eta function converges to the volume of the Bismut-Freed meromorphic family of connection 1-forms. | Adiabatic limits of eta and zeta functions of elliptic operators | 14,584 |
We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the $\alpha$-genus. | Small eigenvalues of the Conformal Laplacian | 14,585 |
Let M be a Riemannian manifold and R its curvature tensor. For a unit vector X tangent to M at a point p, the Jacobi operator is defined by R_X = R(X, .) X$. The manifold M is called pointwise Osserman if, for every point p, the spectrum of the Jacobi operator does not depend of the choice of X, and is called globally Osserman if it depends neither of X, nor of p. R.Osserman conjectured that globally Osserman manifolds are two-point homogeneous. We prove the Osserman Conjecture for dimension n \ne 8, 16, and its pointwise version for n \ne 2,4,8,16. Partial results in the cases n = 8, 16 are also given. | Osserman Conjecture in dimension n \ne 8, 16 | 14,586 |
This note discusses the higher K-energy functionals which were defined by Bando and Mabuchi, and integrate higher Futaki invariants. Two new formulas for the higher K-energy functionals are given, and the second K-energy is shown to be related to Donaldson's Lagrangian applied to metrics on the tangent bundle. | Higher K-energy functionals and higher Futaki invariants | 14,587 |
We study a multiply warped products manifold associated with the Reissner-Nordstrom metric to investigate the physical properties inside the black hole event horizons. It is shown that, different from the uncharged Schwarzschild metric, the Ricci curvature components inside the Reissner-Nordstrom black hole horizons are nonvanishing, while the Einstein scalar curvature vanishes even in the interior of the charged metric. Introducing a perfect fluid inside the Reissor-Nordstrom black hole, it is also shown that the charge plays effective roles of decreasing the mass-energy density and the pressure of the fluid inside the black hole. | Warped products and Reissner-Nordstrom metric | 14,588 |
We shall develop a new deformation theory of geometric structures in terms of closed differential forms. This theory is a generalization of Kodaira -Spencer theory and further we obtain a criterion of unobstructed deformations. We apply this theory to certain geometric structures: Calabi-Yau, HyperK\"ahler, $\G$ and $\Spin$ structures and show that these deformation spaces are smooth in a systematic way. | Deformations of calibrations, {Calabi-Yau, HyperKähler, $G_2$ and
Spin(7) structures} | 14,589 |
Twistor 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 study twistor forms on compact Kaehler manifolds and give a complete description up to special forms in the middle dimension. In particular, we show that they are closely related to Hamiltonian 2-forms. This provides the first examples of compact Kaehler manifolds with non-parallel twistor forms in any even degree. | Twistor Forms on Kaehler Manifolds | 14,590 |
A special K\"ahler-Ricci potential on a K\"ahler manifold is any nonconstant $C^\infty$ function $\tau$ such that $J(\nabla\tau)$ is a Killing vector field and, at every point with $d\tau\ne 0$, all nonzero tangent vectors orthogonal to $\nabla\tau$ and $J(\nabla\tau)$ are eigenvectors of both $\nabla d\tau$ and the Ricci tensor. For instance, this is always the case if $\tau$ is a nonconstant $C^\infty$ function on a K\"ahler manifold $(M,g)$ of complex dimension $m>2$ and the metric $\tilde g=g/\tau^2$, defined wherever $\tau\ne 0$, is Einstein. (When such $\tau$ exists, $(M,g)$ may be called {\it almost-everywhere conformally Einstein}.) We provide a complete classification of compact K\"ahler manifolds with special K\"ahler-Ricci potentials and use it to prove a structure theorem for compact K\"ahler manifolds of any complex dimension $m>2$ which are almost-everywhere conformally Einstein. | Special Kähler-Ricci potentials on compact Kähler manifolds | 14,591 |
We study the solvability of the equation for the smooth function F, H=-k F g, on a geodesically complete pseudo-Riemannian manifold (M,g), H being the covariant Hessian of F. A similar equation was considered by Obata and Gallot in the Riemannian case for positive values of the constant k; the result was that the manifold must be the canonical sphere. In this generalized setting we obtain a range of possibilities, depending on the sign of k, the signature of the metric and the value of a certain first integral of the equation: the manifold is shown to be of constant sectional curvature or a warped product with suitable factors depending on the cases. | Rigidification of pseudo-Riemannian Manifolds by an Elliptic Equation | 14,592 |
This is the third in a series of three papers math.DG/0111324, math.DG/0111326 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using analytic methods. The three papers are surveyed in math.DG/0206016. Let N be such a U(1)-invariant SL 3-fold. Then |z_1|^2-|z_2|^2=2a on N for some real a. Locally, N can be written as a kind of graph of functions u,v : R^2 --> R satisfying a nonlinear Cauchy-Riemann equation depending on a. When a is nonzero, u,v are smooth and N is nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, corresponding to singular points of N. The first paper math.DG/0111324 studied the case a nonzero, and proved existence and uniqueness for solutions of two Dirichlet problems derived from the nonlinear Cauchy-Riemann equation. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C^3, with boundary conditions. The second paper math.DG/0111326 extended these results to weak solutions of the Dirichlet problems when a=0, giving existence and uniqueness of many singular U(1)-invariant SL 3-folds in C^3, with boundary conditions. This third paper studies the singularities of these SL 3-folds. We show that under mild conditions the singularities are isolated, and have a multiplicity n>0, and one of two types. Examples are constructed with every multiplicity and type. We also prove the existence of large families of U(1)-invariant special Lagrangian fibrations of open sets in C^3, including singular fibres. | U(1)-invariant special Lagrangian 3-folds. III. Properties of singular
solutions | 14,593 |
We prove that under certain conditions on the mean curvature and on the Kaehler angles, a compact submanifold M of real dimension 2n, immersed into a Kaehler-Einstein manifold N of complex dimension 2n, must be either a complex or a Lagrangian submanifold of N, or have constant Kaehler angle, depending on n=1, n=2, or n \geq 3, and the sign of the scalar curvature of N. These results generalize to non-minimal submanifolds some known results for minimal submanifolds. Our main tool is a Bochner-type technique involving a formula on the Laplacian of a symmetric function on the Kaehler angles and the Weitzenboeck formula for the Kaehler form of N restricted to M. | On the Kaehler angles of submanifolds | 14,594 |
In this paper, we apply the Tian-Yau-Zelditch expansion of the Bergman kernel on polarized K\"ahler metrics to approximate plurisubharmonic functions and compute the $\alpha$-invariant of $CP^2#2\bar{CP^2}$, which is exactly 1/3. In addition we prove Tian's conjecture on the generalized Moser-Trudinger inequality in a special case prescribed by the $\alpha$-invariant. | The $α$-Invariant on $CP^2#2\bar{CP^2}$ | 14,595 |
We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type $(r,s)$ in a vector space of signature $(p,q)$. We then use these examples to establish some results concerning higher order Osserman and higher order Jordan Osserman algebraic curvature tensors. | The Jordan normal form of higher order Osserman algebraic curvature
tensors | 14,596 |
We show that any $k$ Osserman Lorentzian algebraic curvature tensor has constant sectional curvature and give an elementary proof that any local 2 point homogeneous Lorentzian manifold has constant sectional curvature. We also show that a Szab\'o Lorentzian covariant derivative algebraic curvature tensor vanishes. | Curvature tensors whose Jacobi or Szabo operator is nilpotent on null
vectors | 14,597 |
Let M be a pseudo-Riemannian manifold with a pseudo-Hermitian complex structure $J$. We give necessary and sufficient conditions that the curvature operator $R(\pi)$ is complex linear when $\pi$ is a $J$ invariant real 2 plane. Under this assumption, we study when M is complex IP - i.e. the spectrum, or more generally the Jordan normal form, of $R(\pi)$ is constant on the Grassmannian of complex spacelike or timelike lines. Methods from algebraic topology are used to obtain restrictions on the spectrum of a complex IP algebraic curvature tensor. | Complex IP curvature tensors | 14,598 |
We classify the connected pseudo-Riemannian manifolds of signature $(p,q)$ with $q\ge5$ so that at each point of $M$ the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least one point of $M$. | Algebraic curvature tensors for indefinite metrics whose skew-symmetric
curvature operator has constant Jordan normal form | 14,599 |
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