Showing papers in "Journal of Differential Geometry in 2000"
TL;DR: In this article, the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound was studied.
Abstract: In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound say RicMn i n In Sections and sometimes in we also assume a lower volume bound Vol B pi v In this case the sequence is said to be non collapsing If limi Vol B pi then the sequence is said to collapse It turns out that a convergent sequence is noncollapsing if and only if the limit has positive n dimensional Hausdor measure In par ticular any convergent sequence is either collapsing or noncollapsing Moreover if the sequence is collapsing it turns out that the Hausdor dimension of the limit is actually n see Sections and Our theorems on the in nitesimal structure of limit spaces have equivalent statements in terms of or implications for the structure on a small but de nite scale of manifolds with RicMn n Al though both contexts are signi cant for the most part it is the limit spaces which are emphasized here Typically the relation between corre sponding statements for manifolds and limit spaces follows directly from the continuity of the geometric quantities in question under Gromov Hausdor limits together with Gromov s compactness theorem Theorems see also Remark are examples of
1,031 citations
TL;DR: Aleksandrov as mentioned in this paper showed that the Euclidean space of all smooth Kahler metrics is a path length space of nonpositive curvature in the sense of A. D. Mabuchi.
Abstract: This paper, the second of a series, deals with the function space \mathcal{H} of all smooth Kahler metrics in any given n-dimensional, closed complex manifold V, these metrics being restricted to a given, fixed, real cohomology class, called a polarization of V. This function space is equipped with a pre-Hilbert metric structure introduced by T. Mabuchi [10], who also showed that, formally, this metric has nonpositive curvature. In the first paper of this series [4], the second author showed that the same space is a path length space. He also proved that \mathcal{H} is geodesically convex in the sense that, for any two points of \mathcal{H}, there is a unique geodesic path joining them, which is always length minimizing and of class C1,1. This partially verifies two conjectures of Donaldson [8] on the subject. In the present paper, we show first of all, that the space is, as expected, a path length space of nonpositive curvature in the sense of A. D. Aleksandrov. A second result is related to the theory of extremal Kahler metrics, namely that the gradient flow in \mathcal{H} of the "K energy" of V has the property that it strictly decreases the length of all paths in \mathcal{H}, except those induced by one parameter families of holomorphic automorphisms of M.
558 citations
TL;DR: In this article, the Lp analogues of the Petty projection inequality and the BusemannPetty centroid inequality are established, where the ratio of the functionals is invariant under non-degenerate linear transformations.
Abstract: The Lp analogues of the Petty projection inequality and the BusemannPetty centroid inequality are established. An affine isoperimetric inequality compares two functionals associated with convex (or more general) bodies, where the ratio of the functionals is invariant under non-degenerate linear transformations. These isoperimetric inequalities are more powerful than their better-known Euclidean relatives.
465 citations
TL;DR: In this paper, the deformation theory necessary to obtain virtual moduli cycles of stable sheaves whose higher obstruction groups vanish has been developed, and the moduli spaces of sheaves on a general $K3$ fibration have been computed.
Abstract: We briefly review the formal picture in which a Calabi-Yau $n$-fold is the complex analogue of an oriented real $n$-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of \cite{LT}, \cite{BF} in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in $\Pee^3$, and Donaldson-- and Gromov-Witten-- like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold $X$, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general $K3$ fibration $X$, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the ``Mukai-dual'' 3-fold for others. As an example the invariant is shown to distinguish Gross' diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of $X$.
464 citations
TL;DR: In this article, a diffeomorphism-invariant functional on the space of differential 3-forms on a closed 6-manifold M was introduced, and the functional was restricted to a de Rham cohomology class in H3(M, R).
Abstract: We study the special algebraic properties of alternating 3-forms in 6 dimensions and introduce a diffeomorphism-invariant functional on the space of differential 3-forms on a closed 6-manifold M. Restricting the functional to a de Rham cohomology class in H3(M, R), we find that a critical point which is generic in a suitable sense defines a complex threefold with trivial canonical bundle. This approach gives a direct method of showing that an open set in H3(M, R) is a local moduli space for this structure and introduces in a natural way the special pseudo-Kahler structure on it.
331 citations
TL;DR: The Ricci-flat curvature of the Lagrangian torus fibrations of Calabi-Yau n-folds has been studied in this paper, where it was shown that the curvature can be approximated to O(e−C/∊) for some constant C > 0.
Abstract: Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we make a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kahler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibrations associated with the large complex structure limit, namely an n-sphere, and that the metric on this Sn is induced from a standard (singular) Riemannian metric on the base, the singularities of the metric corresponding to the limit discriminant locus of the fibrations. This conjecture is trivially true for elliptic curves; in this paper we prove it in the case of K3 surfaces. Using the standard description of mirror symmetry for K3 surfaces and the hyperkahler rotation trick, we reduce the problem to that of studying Kahler degenerations of elliptic K3 surfaces, with the Kahler class approaching the wall of the Kahler cone corresponding to the fibration and the volume normalized to be one. Here we are able to write down a remarkably accurate approximation to the Ricci-flat metric: if the elliptic fibres are of area ∊ > 0, then the error is O(e−C/∊) for some constant C > 0. This metric is obtained by gluing together a semi-flat metric on the smooth part of the fibration with suitable Ooguri-Vafa metrics near the singular fibres. For small ∊, this is a sufficiently good approximation that the above conjecture is then an easy consequence.
284 citations
TL;DR: A higher dimensional analogue of Kodaira's canonical bundle formula is obtained in this paper, where it is shown that the log-canonical ring of a klt pair with κ ≤ 3 is finitely generated.
Abstract: A higher dimensional analogue of Kodaira’s canonical bundle formula is obtained. As applications, we prove that the log-canonical ring of a klt pair with κ ≤ 3is finitely generated, and that there exists an effectively computable natural number M such that | MK X | induces the Iitaka fibering for every algebraic threefold X with Kodaira dimension κ =1 .
182 citations
TL;DR: In this article, complete classification results for tight contact structures on two classes of 3-manifolds: torus bundles which fiber over the circle and circle bundles that fiber over closed surfaces.
Abstract: We present complete classification results for tight contact structures on two classes of 3-manifolds: (i) torus bundles which fiber over the circle and (ii) circle bundles which fiber over closed surfaces.
170 citations
TL;DR: In this paper, the authors discuss contravariant connections on Poisson manifolds and show that these connections play an important role in the study of global properties of Poisson manifold, and use them to define Poisson holonomy and new invariants.
Abstract: We discuss contravariant connections on Poisson manifolds. For vector bun dles, the corresponding operational notion of a contravariant derivative had been introduced by I. Vaisman. We show that these connections play an important role in the study of global properties of Poisson manifolds and we use them to define Poisson holonomy and new invariants of Poisson manifolds.
146 citations
TL;DR: In this paper, it was shown that a semialgebraic differentiable mapping has a generalized critical values set of measure zero, and if the mapping is C2, a locally trivial fibration over the complement of this set was obtained.
Abstract: We prove that a semialgebraic differentiable mapping has a generalized critical values set of measure zero. Moreover, if the mapping is C2 we obtain, by a generalisation of Ehresmann's fibration theorem due to P. J. Rabier [20], a locally trivial fibration over the complement of this set. In the complex case, we prove that the set of generalized critical values of a polynomial mapping is a proper algebraic set.
141 citations
Journal Article•
TL;DR: In this paper, the moment maps for hamiltonian quasi-Poisson Lie groups are defined and studied, and an analogue of the hamiltonians reduction theorem for quasi-poisson group actions is shown.
Abstract: A Lie group $G$ in a group pair ($D, G$), integrating the Lie algebra $\mathfrak{g}$ in a Manin pair ($\mathfrak{d,g}$), has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups $G$, and show that they generalize the Poisson actions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are hamiltonian. These moment maps take values in the homogeneous space $D/G$. We prove an analogue of the hamiltonian reduction theorem for quasi-Poisson group actions, and we study the symplectic leaves of the orbit spaces of hamiltonian quasi-Poisson spaces.
TL;DR: In this paper, the authors characterize the topology of 3D Riemannian manifolds with a uniform lower bound of sectional curvature which converges to a metric space of lower dimension.
Abstract: The purpose of this paper is to completely characterize the topology of three-dimensional Riemannian manifolds with a uniform lower bound of sectional curvature which converges to a metric space of lower dimension.
TL;DR: In this article, a generalization of some results on Hilbert schemes of points on surfaces is given, where the moduli spaces of stable bundles on a smooth projective surface are shown to be Gieseker compactified.
Abstract: This paper gives a generalization of some results on Hilbert schemes of points on surfaces. Let M G (r, n) (resp. M U (r, n)) be the Gieseker (resp. Uhlenbeck) compactification of the moduli spaces of stable bundles on a smooth projective surface. We show that, for surfaces satisfying some technical condition:
TL;DR: In this paper, it was shown that a compact two-dimensional Riemannian manifold with strictly convex boundary and with no focal points is deformation boundary rigid, and that two dimensional Anosov manifolds are spectrally rigid.
Abstract: We prove that a compact two-dimensional Riemannian manifold with strictly convex boundary and with no focal points is deformation boundary rigid. We also show, using similar methods, that two-dimensional Anosov manifolds with no focal points are spectrally rigid.
TL;DR: In this paper, the authors studied the spectrum of the Dirac operator on hyperbolic manifolds of finite volume and gave a simple criterion in terms of linking numbers for when essential spectrum can occur.
Abstract: We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S^3 we give a simple criterion in terms of linking numbers for when essential spectrum can occur. We compute the accumulation rate of the eigenvalues of a sequence of closed hyperbolic 2- or 3-manifolds degenerating into a noncompact hyperbolic manifold of finite volume. It turns out that in three dimensions there is no clustering at all.
TL;DR: In this paper, a transversal knot type T K is transversally simple if and only if it is determined by its topological knot type K and its Bennequin number.
Abstract: This paper studies knots that are transversal to the standard contact structure in IR 3 , bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type T K is transversally simple if it is determined by its topological knot type K and its Bennequin number. The main theorem asserts that any T K whose associated K satisfies a condition that we call exchange reducibility is transversally simple. As applications, we give a new proof of a theorem of Eliashberg [El91], which asserts that the unknot is transversally simple, and (with the help of a new theorem of Menasco [Me99]) extend a result of Etnyre [Et99] to prove that all iterated torus knots are transversally simple. We show that the concept of exchange reducibility is the simplest of the constraints that one can place on K in order to prove that any associated T K is transversally simple. We also give examples of pairs of transversal knots that we conjecture are not transversally simple.
TL;DR: In this article, the Lagrangian averaged Navier-Stokes (LANS-α) equations were shown to be well-posed, globally when n = 2, 3 for almost all t in some fixed time interval (0, T) in Hs, s ∊ (n/2 + 1, 3).
Abstract: We establish the existence of three new subgroups of the group of volume-preserving diffeomorphisms of a compact n-dimensional (n ≥ 2) Riemannian manifold which are associated with the Dirichlet, Neumann, and Mixed type boundary conditions that arise in second-order elliptic PDEs. We prove that when endowed with the Hs Hilbert-class topologies for s > (n/2) + 1, these subgroups are C∞ differential manifolds. We consider these new diffeomorphism groups with an H1-equivalent right invariant metric, and prove the existence of unique smooth geodesics η(t, ·) of this metric, as well as existence and uniqueness of the Jacobi equations associated to this metric. Geodesics on these subgroups are, in fact, the flows of a time-dependent velocity vector field u(t, x), so that ∂tη(t, ·) = u(t, η(t, ·)) with η(0, x) = x, and remarkably the vector field u(t, x) solves the so-called Lagrangian averaged Euler (LAE-α) equations on M. These equations, and their viscous counterparts, the Lagrangian averaged Navier-Stokes (LANS-α) equations, model the motion of a fluid at scales larger than an a priori fixed parameter α > 0, while averaging (or filtering-out) the small scale motion, and this is achieved without the use of artificial viscosity. We prove that for divergence-free initial data satisfying u = 0 on ∂M, the LAE-α equations are well-posed, globally when n = 2. We also find the boundary conditions that make the LANS-α equations well-posed, globally when n = 3, and prove that solutions of the LANS-α equations converge when n = 2, 3 for almost all t in some fixed time interval (0, T) in Hs, s ∊ (n/2 + 1, 3) to solutions of the LAE-α equations, thus confirming the scaling arguments of Barenblatt & Chorin.
TL;DR: In this paper, the singular holomorphic curves in a linear system on an algebraic surface were enumerated based on the family Seiberg-Witten invariants and tools from differential topology and algebraic geometry.
Abstract: In this paper, we discuss the scheme of enumerating the singular holomorphic curves in a linear system on an algebraic surface Or approach is based on the usage of the family Seiberg-Witten invariant and tools from differential topology and algebraic geometry In particular, one shows that the number of δ-nodes nodal curves in a generic δ dimensional sub-linear system can be expressed as a universal degree δ polynomial in terms of the four basic numerical invariants of the linear system and the algebraic surface The result enables us to study in detail the structure of these enumerative invariants
TL;DR: In this paper, the author applies the Excess Theorem of Abresch and Gromoll (1990) to prove two theorems: if such a manifold has small linear diameter growth then its fundamental group is finitely generated, and if it has an infinitely generated fundamental group then it has a tangent cone at infinity which is not polar.
Abstract: In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fundamental group is finitely generated. The second states that if such a manifold has an infinitely generated fundamental group then it has a tangent cone at infinity which is not polar. A corollary of either theorem is the fact that if such a manifold has linear volume growth, then its fundamental group is finitely generated.
TL;DR: In this paper, the authors derive adjunction inequalities for embedded surfaces with non-negative self-intersection number in four-manifolds, which are proved by using relations between Seiberg-Witten invariants which are induced from embedded surfaces.
Abstract: In this paper, we derive new adjunction inequalities for embedded surfaces with non-negative self-intersection number in four-manifolds These formulas are proved by using relations between Seiberg-Witten invariants which are induced from embedded surfaces To prove these relations, we develop the relevant parts of a Floer theory for four-manifolds which bound circle-bundles over Riemann surfaces
TL;DR: In this article, it was shown that resolvent of the Laplacian has a meromorphic continuation to a conic neighborhood of the continuous spectrum, and that the number of resonances in such a neighborhood is O(r √ O(n) ).
Abstract: In this paper we discuss meromorphic continuation of the resolvent and bounds on the number of resonances for scattering manifolds, a class of manifolds generalizing Euclidian n-space. Subject to the basic assumption of analyticity near infinity, we show that resolvent of the Laplacian has a meromorphic continuation to a conic neighborhood of the continuous spectrum. This involves a geometric interpretation of the complex scaling method in terms of deformations in the Grauert tube of the manifold. We then show that the number of resonances (poles of the meromorphic continuation of the resolvent) in a conic neighborhood of $\mathbb{R}_+$of absolute value less than $r^2$ is $\mathcal O(r^n)$. Under the stronger assumption of global analyticity and hyperbolicity of the geodesic flow, we prove a finer, Weyl-type upper bound for the counting function for resonances in small neighborhoods of the real axis. This estimate has an exponent which involves the dimension of the trapped set of the geodesic flow.
TL;DR: In this article, the construction of previously unknown fundamental groups for positively curved manifolds was studied, and the fundamental groups were shown to be monotonically convex and non-convex.
Abstract: This paper deals with the construction of previously unknown fundamental groups for positively curved manifolds.
TL;DR: In this article, the moduli space of holomorphic semistable principal G-bundles over an elliptic curve is constructed by considering deformations of a minimally unstable G bundle, and the set of all such deformations can be described as the C^* quotient of the cohomology group of a sheaf of unipotent groups.
Abstract: This paper continues the study of holomorphic semistable principal G-bundles over an elliptic curve In this paper, the moduli space of all such bundles is constructed by considering deformations of a minimally unstable G-bundle The set of all such deformations can be described as the C^* quotient of the cohomology group of a sheaf of unipotent groups, and we show that this quotient has the structure of a weighted projective space We identify this weighted projective space with the moduli space of semistable G-bundles, giving a new proof of a theorem of Looijenga
TL;DR: In this article, the existence of contact submanifolds realizing the Poincare dual of the top Chern class of a complex vector bundle over a closed contact manifold is proved.
Abstract: We prove the existence of contact submanifolds realizing the Poincare dual of the top Chern class of a complex vector bundle over a closed contact manifold. This result is analogue in the contact categoryto Donaldson's construction of symplectic submanifolds. The main tool in the construction is to show the existence of sequences of sections which are asymptotically holomorphic in an appropiate sense and that satisfya transversalitywith es- timates propertydirectlyin the contact category . The description of the ob- tained contact submanifolds allows us to prove an extension of the Lefschetz hyperplane theorem which completes their topological characterization.
TL;DR: In this article, it was shown that the higher signatures of a compact oriented manifold with boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic are oriented-homotopy invariants.
Abstract: If M is a compact oriented manifold-with-boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic, we show that the higher signatures of M are oriented-homotopy invariants. We give applications to the question of when higher signatures of closed manifolds are cut-and-paste invariant.