Let @math be a contact form on a manifold @math , and @math a closed Legendrian submanifold. I prove that @math intersects some characteristic for @math at least twice if all characteristics are closed and of the same period, and @math embeds nicely into the product of @math and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most @math . This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.
Let @math be a bounded star-shaped domain with smooth boundary and @math a closed Legendrian submanifold of nonpositive curvature. The last condition means that @math that admits a Riemannian metric of nonpositive sectional curvature. In the recent preprint @cite_20 K. Cieliebak and K. Mohnke proved that @math possesses a Reeb chord of length bounded above by the (toroidal) Lagrangian capacity of @math , see [Corollary 1.12] CM . Using [Corollary 1.3] CM , they deduced that @math admits a Reeb chord of length bounded above by @math , if @math , and @math , i.e., @math is the unit sphere.
@cite_20
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We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdor limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X;d) in which the distance between two points equals the inmum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having bounded below by K", with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the GromovHausdor topology on compact metric spaces (modulo isometries); they form Let @math be a geometrically bounded symplectic manifold, @math be a closed, regular coisotropic submanifold, and @math be a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of @math is bounded below by the sum of the Betti numbers of @math , provided that the Hofer distance between @math and the identity is small enough and the pair @math is non-degenerate. As an application, I prove a presymplectic non-embedding result. A version of the Arnold-Givental conjecture for coisotropic submanifolds is also discussed. Finite convex geometries are combinatorial structures. It follows from a recent result of M. Richter and L.G. Rogers that there is an infinite set @math of planar convex polygons such that @math with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of @math to a finite subset in a natural way. An @math is a differentiable convex simple closed curve @math in the plane having an inscribed circle of radius @math and a circumscribed circle of radius @math such that the ratio @math is at least @math . Motivated by Richter and Rogers' result, we construct a set @math such that (1) @math contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) @math with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to @math , @math is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive @math and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets @math of @math such that each @math consists of almost-circles of accuracy @math and the convex geometry in question is represented by restricting the convex hull operator to @math . The affine-disjointness of @math and @math means that, in addition to @math , even @math is disjoint from @math for every non-degenerate affine transformation @math . Let M be a closed three-dimensional manifold with contact form λ so that Ker λ is tight. In this paper, we will present a first application of the filling method by pseudoholomorphic curves recently developed by the author. We will show that Legendrian knots L ⊂ M satisfying suitable assumptions admit a Reeb Chord, that is, there is a trajectory x(t) of the Reeb vector field and T > 0 such that x(0),x(T) ∈ L and x(0) ≠ x(T). In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of ( )-numbers. These ( )-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones’ result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones’ ( )-numbers. This variant has since been used by others to give structure theorems for rectifiable sets and to give upper bounds for the measure of a set. In this paper we give a version of P. Jones’ theorem for sets of arbitrary (integer) dimension lying in Euclidean space. Our main result is a lower bound for the d-dimensional Hausdorff measure of a set in terms of an analogous sum of ( )-type numbers. We also show an upper bound of this type. The combination of these results gives a Jones theorem for higher dimensional sets. While there is no assumption of Ahlfors regularity, or of a measure on the underlying set, there is an assumption of a lower bound on the Hausdorff content. We adapt David and Semmes’ version of Jones’ ( )-numbers by redefining them using a Choquet integral, allowing them to be defined for arbitrary sets (and not just sets of locally finite measure). A key tool in the proof is G. David and T. Toro’s parametrization of Reifenberg flat sets (with holes). We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle, acting as reparameterizations. In particular we investigate the L^2 inner product with respect to 1 plus curvature squared times arclength as the measure along a curve, applied to normal vector field to the curve. The curvature squared term acts as a sort of geometric Tikhonov regularization because, without it, the geodesic distance between any 2 distinct curves is 0, while in our case the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are positive or 0, while for curves with high curvature or perturbations of high frequency, the curvatures are negative. Abstract We introduce and study rough (approximate) lower curvature bounds for discrete spaces and for graphs. This notion agrees with the one introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press] and [K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65–131], in the sense that the metric measure space which is approximated by a sequence of discrete spaces with rough curvature ⩾ K will have curvature ⩾ K in the sense of [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65–131]. Moreover, in the converse direction, discretizations of metric measure spaces with curvature ⩾ K will have rough curvature ⩾ K . We apply our results to concrete examples of homogeneous planar graphs. Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. Let (S, fλ) be the three sphere endowed with a contact form fλ, where λ is the standard tight contact form on S and f : S → R 0 is a smooth function. Moreover let L ⊂ S be a Legendrian knot. Then V.I. Arnold raised the conjecture in 1986 that there is a characteristic chord, i.e. an orbit x of the Reeb vectorfield that intersects the knot L at two different times. Characteristic chords are important in the theory of contact homology, see [6], [5]. Very little is known about their existence or non-existence . Using a deep result by H. Hofer, K. Wysocki and E. Zehnder about the existence of a global surface of section in strictly convex hypersurfaces in R, we observe that characteristic chords exist for certain Legendrian knots in strictly convex hypersurfaces in R. We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant λ > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain—with boundary values −λ on one boundary arc and λ on the complementary arc—the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are −a 0 on the complementary arc, then the convergence is to SLE(4; a λ - 1, b λ - 1), a variant of SLE(4). Let (Mn, g) be a compact Riemannian manifold with boundary and dimensionn⩾2. In this paper we discuss the first non-zero eigenvalue problem (1) Problem (1) is known as the Stekloff problem because it was introduced by him in 1902, for bounded domains of the plane. We discuss estimates of the eigenvalueν1in terms of the geometry of the manifold (Mn, g). In the two-dimensional case we generalize Payne's Theorem [P] for bounded domains in the plane to non-negative curvature manifolds. In this case we show thatν1⩾k0, wherekg⩾k0andkgrepresents the geodesic curvature of the boundary. In higher dimensionsn⩾3 for non-negative Ricci curvature manifolds we show thatν1>k0 2, wherek0is a lower bound for any eigenvalue of the second fundamental form of the boundary. We introduce an isoperimetric constant and prove a Cheeger's type inequality for the Stekloff eigenvalue. Let (L ) be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold ((X , ) ). We assume that the first homology group (H_1 (L ; A) ) with coefficients in a commutative ring (A ) injects into the group (H_1 (X ; A) ) and that (X ) contains no Maslov zero pseudo-holomorphic disc with boundary on (L ). Then, we prove that for every generic choice of a tame almost-complex structure (J ) on (X ), every relative homology class (d H_2 (X , L ; Z ) ) and adequate number of incidence conditions in (L ) or (X ), the weighted number of (J )-holomorphic discs with boundary on (L ), homologous to (d ), and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of (J ), provided that at least one incidence condition lies in (L ). These numbers thus define open Gromov–Witten invariants in dimension six, taking values in the ring (A ). We consider the problem of covering and packing subsets ofΔ-hyperbolic metric spaces and graphs by balls.These spaces, defined via a combinatorial Gromov condition, haverecently become of interest in several domains of computer science.Specifically, given a subset Sof aΔ-hyperbolic graph Gand a positive numberR, let Δ(S,R) be theminimum number of balls of radius Rcovering S.It is known that computing Δ(S,R)or approximating this number within a constant factor is hard evenfor 2-hyperbolic graphs. In this paper, using a primal-dualapproach, we show how to construct in polynomial time a covering ofSwith at most Δ(S,R)balls of (slightly larger) radius R+ Δ.This result is established in the general framework ofΔ-hyperbolic geodesic metric spaces and is extendedto some other set families derived from balls. The coveringalgorithm is used to design better approximation algorithms for theaugmentation problem with diameter constraints and for thek-center problem in Δ-hyperbolicgraphs. An (α,β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v, H) ≤ α·dist(u, v, G) + β for every pair (u, v) ∈ V ×V, where dist(u,v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1,0)-spanner of size O(n 1 + 1 k ). This size-stretch bound is essentially optimal by the girth conjecture. Yet, it is important to note that any argument based on the girth only applies to adjacent vertices. It is therefore intriguing to ask if one can “bypass” the conjecture by settling for a multiplicative stretch of 2k − 1 only for neighboring vertex pairs, while maintaining a strictly better multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of k-hybrid spanners, in which non neighboring vertex pairs enjoy a multiplicative k stretch and the neighboring vertex pairs enjoy a multiplicative (2k − 1) stretch (hence, tight by the conjecture). We show that for every unweighted n-vertex graph G, there is a (polynomially constructible) k-hybrid spanner with O(k 2 ·n 1 + 1 k ) edges. This should be compared against the current best (α,β) spanner construction of [5] that obtains (k,k − 1) stretch with O(k ·n 1 + 1 k ) edges. An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs S ×V for a given subset of vertices S ⊆ V referred to here as sources. Spanners in which the distances in S ×V are bounded are referred to as sourcewise spanners. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more). This book consists of two independent works: Part I is 'Solutions of the Einstein Vacuum Equations', by Lydia Bieri. Part II is 'Solutions of the Einstein-Maxwell Equations', by Nina Zipser. A famous result of Christodoulou and Klainerman is the global nonlinear stability of Minkowski spacetime. In this book, Bieri and Zipser provide two extensions to this result. In the first part, Bieri solves the Cauchy problem for the Einstein vacuum equations with more general, asymptotically flat initial data, and describes precisely the asymptotic behavior. In particular, she assumes less decay in the power of @math and one less derivative than in the Christodoulou-Klainerman result. She proves that in this case, too, the initial data, being globally close to the trivial data, yields a solution which is a complete spacetime, tending to the Minkowski spacetime at infinity along any geodesic. In contrast to the original situation, certain estimates in this proof are borderline in view of decay, indicating that the conditions in the main theorem on the decay at infinity on the initial data are sharp. In the second part, Zipser proves the existence of smooth, global solutions to the Einstein-Maxwell equations. A nontrivial solution of these equations is a curved spacetime with an electromagnetic field. To prove the existence of solutions to the Einstein-Maxwell equations, Zipser follows the argument and methodology introduced by Christodoulou and Klainerman. To generalize the original results, she needs to contend with the additional curvature terms that arise due to the presence of the electromagnetic field @math ; in her case the Ricci curvature of the spacetime is not identically zero but rather represented by a quadratic in the components of @math . In particular the Ricci curvature is a constant multiple of the stress-energy tensor for @math . Furthermore, the traceless part of the Riemann curvature tensor no longer satisfies the homogeneous Bianchi equations but rather inhomogeneous equations including components of the spacetime Ricci curvature. Therefore, the second part of this book focuses primarily on the derivation of estimates for the new terms that arise due to the presence of the electromagnetic field. Let @math be a compact Riemannian submanifold of @math of dimension @math and let @math be a sample of i.i.d. points in @math with uniform distribution. We study the random operators @math where @math is the Gaussian kernel and @math as @math Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of @math @math (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations @math for smooth functions @math both pointwise and uniformly in @math and @math (here @math and @math is the Riemannian volume measure). In particular, we show that for any class @math of three times differentiable functions on @math with uniformly bounded derivatives @math as soon as @math and also prove asymptotic normality of @math (functional CLT) for a fixed @math and uniformly in @math Arnold conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a characteristic chord with respect to any contact form. I confirm this conjecture if the know has Thurston-Bennequin invariant −1. More generally, existence of chords is proved for a standard Legendrian unknot on the boundary of a subcritical Stein manifold of any dimension. There is also a multiplicity result which implies in some situations existence of infinitely many chords.¶The proof relies on the behaviour of symplectic homology under handle attaching. The main observation is that symplectic homology only changes in the presence of chords. A standard result in metric geometry is that every compact geodesic metric space can be approximated arbitrarily well by finite metric graphs in the Gromov-Hausdorff sense. It is well known that the first Betti number of the approximating graphs may blow up as the approximation gets finer. In our work, given a compact geodesic metric space @math , we define a sequence @math of non-negative real numbers by @math By construction, and the above result, this is a non-increasing sequence with limit @math . We study this sequence and its rates of decay with @math . We also identify a precise relationship between the sequence and the first Vietoris-Rips persistence barcode of @math . Furthermore, we specifically analyze @math and find upper and lower bounds based on hyperbolicity and other metric invariants. As a consequence of the tools we develop, our work also provides a Gromov-Hausdorff stability result for the Reeb construction on geodesic metric spaces with respect to the function given by distance to a reference point. In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ℝ d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ℬ R ≈(ℒ−λ I)−1, where ℒ is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ℝ d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e −i κ‖x‖ ‖x‖, x∈ℝ d , κ∈ℝ+, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ |log e|2 n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2), even in the case κ=O(n). We set up an algebraic framework for the study of pseudoholomorphic discs bounding nonorientable Lagrangians, as well as equivariant extensions of such structures arising from a torus action. First, we define unital cyclic twisted @math algebras and prove some basic results about them, including a homological perturbation lemma which allows one to construct minimal models of such algebras. We then construct an equivariant extension of @math algebras which are invariant under a torus action on the underlying complex. Finally, we construct a homotopy retraction of the Cartan-Weil complex to equivariant cohomology, which allows us to construct minimal models for equivariant cyclic twisted @math algebras. In a forthcoming paper we will use these results to define and obtain fixed-point expressions for the open Gromov-Witten theory of @math , as well as its equivariant extension.