Showing papers in "Journal of Differential Geometry in 2005"
TL;DR: In this article, an interpretation of the Kervaire invariant of a Riemannian manifold in terms of a holomorphic line bundle on the abelian variety was presented.
Abstract: We describe an interpretation of the Kervaire invariant of a Riemannian manifold of dimension $4k+2$ in terms of a holomorphic line bundle on the abelian variety $H^{2k+1}(M)\otimes R/Z$. Our results are inspired by work of Witten on the fivebrane partition function in $M$-theory (hep-th/9610234, hep-th/9609122). Our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics, and is based on our theory of differential functions. These differential functions generalize the differential characters of Cheeger-Simons, and the bulk of this paper is devoted to their study.
344 citations
TL;DR: In this article, it was shown that test configurations give new lower bounds on the curvature of the scalar curvature on a Riemannian manifold, using asymptotic approximation by finite-dimensional problems.
Abstract: The main result of this paper shows that "test configurations" give new lower bounds on the $L^{2}$ norm of the scalar curvature on a Kahler manifold. This is closely analogous to the analysis of the Yang-Mills functional over Riemann surfaces by Atiyah and Bott. The proof uses asymptotic approximation by finite-dimensional problems: the essential ingredient being the Tian-Zelditch-Lu expansion of the "density of states" function.
286 citations
TL;DR: In this article, the convergence of the Yamabe flow was shown to hold if the dimension of the initial metric is locally conformally flat and the curvature of the scalar curvature is known.
Abstract: We consider the Yamabe flow $\frac{\partial g}{\partial t} = -(R_g - r_g) \, g$ where $g$is a Riemannian metric on a compact manifold $M, R_g$ denotes its scalar curvature, and $r_g$ denotes the mean value of the scalar curvature. We prove convergence of the Yamabe flow if the dimension $n$ satisfies $3 \leq n \leq 5$ or the initial metric is locally conformally flat.
199 citations
TL;DR: In this article, the authors show that the partition function of Chern-Simons theory admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on a Seifert manifold.
Abstract: We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface Σ). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.
192 citations
TL;DR: In this paper, the authors derived an existence criterion to the Supersymmetric String Theory with Torsion proposed by Strominger and proved the existence of such theory for a class of Calabi-Yau three-folds.
Abstract: We derived an existence criterion to the Supersymmetric String Theory with Torsion proposed by Strominger and proved the existence of such theory for a class of Calabi-Yau threefolds.
184 citations
TL;DR: In this article, the authors define the contact homology for Legendrian submanifolds in standard contact (2n + 1)-space using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space.
Abstract: We define the contact homology for Legendrian submanifolds in standard contact (2n + 1)-space using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. This homology provides new invariants of Legendrian isotopy which indicate that the theory of Legendrian isotopy is very rich. Indeed, in [4], the homology is used to detect infinite families of pairwise non-isotopic Legendrian submanifolds which are indistinguishable using previously known invariants.
184 citations
TL;DR: In the standard contact (2n + 1)-space when n > 1, the authors constructed infinite families of pairwise non-Legendrian isotopic, Legendrian n-spheres, n-tori and surfaces which are indistinguishable using classically known invariants.
Abstract: In the standard contact (2n + 1)-space when n > 1, we construct infinite families of pairwise non-Legendrian isotopic, Legendrian n-spheres, n-tori and surfaces which are indistinguishable using classically known invariants. When n is even, these are the first known examples of non-Legendrian isotopic, Legendrian submanifolds of (2n + 1)-space. Such constructions indicate a rich theory of Legendrian submanifolds. To distinguish our examples, we compute their contact homology which was rigorously defined in this situation in [7].
154 citations
TL;DR: In this paper, the authors prove a priori estimates for solutions to the Yamabe problem given a compact Riemannian manifold (Mn, g), with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere.
Abstract: Given a compact Riemannian manifold (Mn, g), with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem We restrict ourselves to the dimensions where the Positive Mass Theorem is known to be true, that is, when n ≤ 7 We also show that, when n ≥ 6, the Weyl tensor has to vanish at a point where solutions to the Yamabe equation blow up
131 citations
TL;DR: In this article, it was shown that conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.
Abstract: We show that C2 conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.
127 citations
TL;DR: In this article, the existence of a universal constant $C$ such that the right-hand side of (1) is smaller than a geometric constant, Σ is W2,2 close to a round sphere was shown.
Abstract: Let $\Sigma \in \mathbf{R}^3$ be a smooth compact connected surface without boundary and denote by $A$ its second fundamental form. We prove the existence of a universal constant $C$ such that \begin{equation} \inf_{\lambda\in {\bf R}}\Vert A - \lambda \rm{Id} \Vert_{L^2(\Sigma)} \leq C \Vert A - \frac{\rm{tr}A}{2} \rm {Id} \Vert_{L^2(\Sigma)^\cdot} \end{equation} Building on this, we also show that, if the right-hand side of (1) is smaller than a geometric constant, Σ is W2,2–close to a round sphere.
122 citations
TL;DR: In this article, it was shown that the bottom of the spectrum for the Laplacian is bounded from above by m2 for a complete, m-dimensional, Kahler manifold with holomorphic bisectional curvature bounded from below by −1.
Abstract: The first part of this paper is devoted to proving a comparison theorem for Kahler manifolds with holomorphic bisectional curvature bounded from below. The model spaces being compared to are ℙℂm, ℙm, and ℙℍm. In particular, it follows that the bottom of the spectrum for the Laplacian is bounded from above by m2 for a complete, m-dimensional, Kahler manifold with holomorphic bisectional curvature bounded from below by −1. The second part of the paper is to show that if this upper bound is achieved and when m=2, then it must have at most four ends.
TL;DR: In this paper, local holomorphic mappings between classical domains in complex projective spaces of different dimensions were studied and it was shown that these mappings possess strong super-rigidity properties when the hyperquadrics have positive signatures.
Abstract: We study local holomorphic mappings sending a piece of a real hyperquadric in a complex space into a hyperquadric in another complex space of possibly larger dimension We show that these mappings possess strong super-rigidity properties when the hyperquadrics have positive signatures These results are applied in the context of holomorphic mappings between classical domains in complex projective spaces of different dimensions
TL;DR: In this article, the authors studied the group of autoequivalences of derived categories of coherent sheaves on the minimal resolution of An-singularities on surfaces, and the main result is to find generators of this group.
Abstract: In this article, we study the group of autoequivalences of derived categories of coherent sheaves on the minimal resolution of An-singularities on surfaces. Our main result is to find generators of this group.
TL;DR: In this article, a closed hyperbolic n-manifold M is studied and the geometry of those maximal spacetimes in terms of cosmological time is studied.
Abstract: Given a closed hyperbolic n-manifold M, we study the flat Lorentzian structures on M×ℝ such that M×{0} is a Cauchy surface. We show there exist only two maximal structures sharing a fixed holonomy (one future complete and the other one past complete). We study the geometry of those maximal spacetimes in terms of cosmological time. In particular, we study the asymptotic behaviour of the level surfaces of the cosmological time. As a by-product, we get that no affine deformation of the hyperbolic holonomy ρ: π1(M)→ SO(n,1) of M acts freely and properly on the whole Minkowski space. The present work generalizes the case n=2 treated by Mess, taking from a work of Benedetti and Guadagnini the emphasis on the fundamental role played by the cosmological time. In the last sections, we introduce measured geodesic stratifications on M, that in a sense furnish a good generalization of measured geodesic laminations in any dimension and we investigate relationships between measured stratifications on M and Lorentzian structures on M×ℝ.
TL;DR: The uniqueness of diffeomorphism type of CP2 ∖ D, where D is a pseudo holomorphic rational curve with one (2, 3)-cusp, was proved in this article.
Abstract: It is proved that the diffeomorphism type of the minimal symplectic fillings of the link of a simple singularity is unique. In the proof, the uniqueness of the diffeomorphism type of CP2 ∖ D, where D is a pseudo holomorphic rational curve with one (2,3)-cusp, is discussed.
TL;DR: In this paper, it was shown that there is a twistor-like correspondence between Finsler metrics on ΩPn whose geodesics are projective lines and a class of symplectic forms on the Grassmannian of 2-planes in ℩n+1.
Abstract: Inspired by Hofer's definition of a metric on the space of compactly supported Hamiltonian maps on a symplectic manifold, this paper exhibits an area-length duality between a class of metric spaces and a class of symplectic manifolds. Using this duality, it is shown that there is a twistor-like correspondence between Finsler metrics on ℝPn whose geodesics are projective lines and a class of symplectic forms on the Grassmannian of 2-planes in ℝn+1.
TL;DR: In this article, the authors studied discrete group actions on coarse Poincare duality spaces, eg, acyclic simplicial complexes which admit free cocompact group actions by poincare-duality groups.
Abstract: We study discrete group actions on coarse Poincare duality spaces, eg, acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups When G is an (n−1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action G ↷ X determines a collection of “peripheral” subgroups H1, … Hk ⊂ G so that the group pair (G, {H1,…Hk }) is an n-dimensional Poincare duality pair In particular, if G is a 2-dimensional 1-ended group of type FP2, and G ↷ X is a free simplicial action on a coarse PD(3) space X, then G contains surface subgroups; if in addition X is simply connected, then we obtain a partial generalization of the Scott/Shalen compact core theorem to the setting of coarse PD(3) spaces In the process, we develop coarse topological language and a formulation of coarse Alexander duality which is suitable for applications involving quasi-isometries and geometric group theory
TL;DR: In this paper, generalized Donaldson invariants of 4-manifolds are defined, using moduli spaces of anti-self-dual connections with structure group SU(N) or PSU(N).
Abstract: Generalized Donaldson invariants of 4-manifolds are defined, using moduli spaces of anti-self-dual connections with structure group SU(N) or PSU(N). Some values of the invariants are calculated for the case that the 4-manifold arises by the knot-complement construction of Fintushel and Stern. The results are consistent with predictions of Marino and Moore.
TL;DR: In this paper, it was shown that closed strictly convex surfaces contracting with normal velocity equal to |A|2 shrink to a point in finite time, after appropriate rescaling, they converge to spheres.
Abstract: We show that closed strictly convex surfaces contracting with normal velocity equal to |A|2 shrink to a point in finite time. After appropriate rescaling, they converge to spheres. We describe our algorithm to find the main test function.
TL;DR: In this paper, the authors proved the existence of a solution to the Monge-Ampere equation on a cone over a thrice-punctured two-sphere.
Abstract: We prove the existence of a solution to the Monge-Ampere equation
detHess(o) = 1 on a cone over a thrice-punctured two-sphere. The total
space of the tangent bundle is thereby a Calabi-Yau manifold with flat special Lagrangian fibers. (Each fiber can
be quotiented to three-torus if the affine monodromy can be shown to lie in SL(3,Z) × R3.) Our method is
through Baues and Cortes's result that a metric cone over an elliptic affine sphere has a parabolic
affine sphere structure (i.e., has a Monge-Ampere solution). The elliptic affine
sphere structure is determined by a semilinear PDE on CP1 minus three points, and we prove existence of a solution
using the direct method in the calculus of variations.
TL;DR: In this article, the authors studied the local deformation space of 3-dimensional cone-manifold structures of constant curvature and showed that the singular locus will be a trivalent graph.
Abstract: We study the local deformation space of 3-dimensional cone-manifold structures of constant curvature κ is an element of {.1, 0, 1} and cone-angles ≤ π. Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main result is a vanishing theorem for the first L2-cohomology group of the smooth part of the cone-manifold with coefficients in the flat bundle of infinitesimal isometries. We conclude local rigidity from this. In the Euclidean case we prove that the first L2-cohomology group of the smooth part with coefficients in the flat tangent bundle is represented by parallel forms.
TL;DR: In this article, the trilinear period on automorphic representations of PG L2(R) is estimated using the analytic structure of the corresponding unique trinear functional on unitary representations.
Abstract: We describe a new method to estimate the trilinear period on automorphic representations of PG L2(R). Such a period gives rise to a special value of the triple L-function. We prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value. Our method is based on the study of the analytic structure of the corresponding unique trilinear functional on unitary representations of PG L2(R).
TL;DR: In this paper, it was shown that if qi, q'i ≥ 2 for all 1 ≤ i ≤ r, then any irreducible subvariety X of Gr(m, n) whose homology class is an integral multiple of the Schubert class [σa] of type a is a SchUbert variety of type σ.
Abstract: Let a = (pq11, . . . , pqrr)
be a partition and a' = (p'1q'1, . . . , p'rq'r)
be its conjugate. We will prove that if qi, q'i ≥ 2 for all 1 ≤ i ≤ r,
then any irreducible subvariety X of Gr(m, n) whose homology class is an integral multiple of the
Schubert class [σa] of type a is a Schubert variety of type a.
TL;DR: In this paper, it was shown that there is an integral homology class in a Kahler-Einstein surface that can be represented by a lagrangian two-sphere, but that a minimizer of area among lagrangians representing this class has isolated singularities with non-flat tangent cones.
Abstract: We show that there is an integral homology class in a Kahler-Einstein surface that can be represented by a lagrangian two-sphere, but that a minimizer of area among lagrangian two-spheres representing this class has isolated singularities with non-flat tangent cones.
TL;DR: In this paper, it was shown that in each dimension n ≥ 10 there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian n -manifolds with 0 ≤ sec ≤ 1, positive Ricci curvature and uniformly bounded diameter.
Abstract: We show that in each dimension n ≥ 10 there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian n -manifolds with 0 ≤ sec ≤ 1, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non-homeomorphic. Finally, we construct examples of noncompact manifolds whose moduli spaces of complete metrics with sec ≥ 0 have infinitely many connected components.
TL;DR: In this article, it was shown that K is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in (9, 10, 15).
Abstract: Let K be the space of properly embedded minimal tori in quotients of R 3 by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that K is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in (9, 10, 15). The degenerate limits of surfaces in K are the catenoid, the helicoid and three 1-parameter families of surfaces: the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examples.
TL;DR: In this article, the notion of moment map for the Hamiltonian action of a Lie group on a symplectic manifold is shown to be a special case of a much more general notion.
Abstract: In this paper, we show that the notion of moment map for the Hamiltonian action of a Lie group on a symplectic manifold is a special case of a much more general notion. In particular, we show that one can associate a moment map to a family of Hamiltonian symplectomorphisms, and we prove that its image is characterized, as in the classical case, by a generalized “energy-period” relation.
TL;DR: In this article, the canonical Chow quotient of a smooth projective variety by a reductive algebraic group is studied, and the main purpose is to introduce the Perturbation-Translation-Specialization relation that gives a computable characterization of the Chow cycles of Chow quotients.
Abstract: This paper studies the canonical Chow quotient of a smooth projective variety by a reductive algebraic group. The main purpose is to introduce the Perturbation–Translation–Specialization relation that gives a computable characterization of the Chow cycles of the Chow quotient. Also, we provide, in the languages that are familiar to topologists and differential geometers, many topological interpretations of Chow quotient that have the advantage to be more intuitive and geometric. More precisely, over the field of complex numbers, these interpretations are, symplectically, the moduli spaces of stable orbits with prescribed momentum charges; and topologically, the moduli space of stable action-manifolds.
TL;DR: In this paper, a family of examples of elliptic curves over number fields whose p-power Selmer groups grow systematically in Z d-extensions is presented, and a somewhat different exposition and proof of Nekova˘ r's theorem is given.
Abstract: In this paper, we show how to use a recent theorem of Nekova˘ r (12) to produce families of examples of elliptic curves over number fields whose p-power Selmer groups grow systematically in Z d- extensions. We give a somewhat different exposition and proof of Nekova˘ r's theorem, and we show in many cases how to replace the fundamental requirement that the elliptic curve has odd p-Selmer rank by a root number calculation.
TL;DR: In this paper, the integer valued SU(3) Casson in-variant defined in (6) was computed by resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the per-turbed flat modulus space.
Abstract: We develop techniques for computing the integer valued SU(3) Casson in- variant defined in (6). Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the per- turbed flat moduli space. These techniques, together with Bott-Morse theory and the splitting principle for spectral flow, are applied to calculateSU(3)(�) for all Brieskorn homology spheres.