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1.
Facundo Mémoli 《Foundations of Computational Mathematics》2011,11(4):417-487
This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling
and tackling the practical problems of object matching and comparison. Objects are viewed as metric measure spaces, and based
on ideas from mass transportation, a Gromov–Wasserstein type of distance between objects is defined. This reformulation yields
a distance between objects which is more amenable to practical computations but retains all the desirable theoretical underpinnings.
The theoretical properties of this new notion of distance are studied, and it is established that it provides a strict metric
on the collection of isomorphism classes of metric measure spaces. Furthermore, the topology generated by this metric is studied,
and sufficient conditions for the pre-compactness of families of metric measure spaces are identified. A second goal of this
paper is to establish links to several other practical methods proposed in the literature for comparing/matching shapes in
precise terms. This is done by proving explicit lower bounds for the proposed distance that involve many of the invariants
previously reported by researchers. These lower bounds can be computed in polynomial time. The numerical implementations of
the ideas are discussed and computational examples are presented. 相似文献
2.
In the Riemannian case, our approach to warped products illuminates curvature formulas that previously seemed formal and somewhat
mysterious. Moreover, the geometric approach allows us to study warped products in a much more general class of spaces. For
complete metric spaces, it is known that nonpositive curvature in the Alexandrov sense is preserved by gluing on isometric
closed convex subsets and by Gromov–Hausdorff limits with strictly positive convexity radius; we show it is also preserved
by warped products with convex warping functions.
Received: 9 January 1998/ Revised version: 12 March 1998 相似文献
3.
Gregory J. Pearlstein 《manuscripta mathematica》2000,102(3):269-310
Following C. Simpson, we show that every variation of graded-polarized mixed Hodge structure defined over ℚ carries a natural
Higgs bundle structure which is invariant under the ℂ* action studied in [20]. We then specialize our construction to the context of [6], and show that the resulting Higgs field
θ determines (and is determined by) the Gromov–Witten potential of the underlying family of Calabi–Yau threefolds.
Received: 14 February 2000 相似文献
4.
One of our two main results exhibits for a vector bundle over a compact Hausdorff space X an interplay between its span, its possible splittings, and the Lyusternik–Shnirel'man category of X. The other main result, also on vector bundles and the Lyusternik–Shnirel'man category, enables us to derive certain inequalities
connecting the immersion codimension, the stable span, and the Lyusternik–Shnirel'man category of a smooth closed manifold
which is not stably parallelizable. Our results are applicable in various situations of general interest.
Received: 4 September 1997 相似文献
5.
We construct random locally compact real trees called Lévy trees that are the genealogical trees associated with continuous-state
branching processes. More precisely, we define a growing family of discrete Galton–Watson trees with i.i.d. exponential branch
lengths that is consistent under Bernoulli percolation on leaves; we define the Lévy tree as the limit of this growing family
with respect to the Gromov–Hausdorff topology on metric spaces. This elementary approach notably includes supercritical trees
and does not make use of the height process introduced by Le Gall and Le Jan to code the genealogy of (sub)critical continuous-state
branching processes. We construct the mass measure of Lévy trees and we give a decomposition along the ancestral subtree of
a Poisson sampling directed by the mass measure.
T. Duquesne is supported by NSF Grants DMS-0203066 and DMS-0405779. M. Winkel is supported by Aon and the Institute of Actuaries,
EPSRC Grant GR/T26368/01, le département de mathématique de l’Université d’Orsay and NSF Grant DMS-0405779. 相似文献
6.
We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive
sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic
or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions.
In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the
Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic.
Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one
containing flats. By the same methods we get an explicit resolution of the singularities in the Baily–Borel resp.Siu–Yau compactification
of finite volume quotients of the complex hyperbolic space.
Oblatum 2-IX-1994 & 7-VIII-1995 相似文献
7.
Jean-François Le Gall 《Inventiones Mathematicae》2007,169(3):621-670
We discuss scaling limits of large bipartite planar maps. If p≥2 is a fixed integer, we consider, for every integer n≥2, a random planar map M
n
which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of M
n
, equipped with the graph distance rescaled by the factor n
-1/4, converges in distribution as n→∞ towards a limiting random compact metric space, in the sense of the Gromov–Hausdorff distance. We prove that the topology
of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence
relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the
limit is almost surely equal to 4. 相似文献
8.
For an arbitrary fibre bundle with a connection, the holonomy group of which is a Lie transformation group, it is shown how
the parallel displacement along a null-homotopic loop can be obtained from the curvature by integration. The result also sheds
some new light on the situation for vector bundles and principal fibre bundles. The Theorem of Ambrose–Singer is derived as
a corollary in our general setting. The curvature of the connection is interpreted as a differential 2-form with values in
the holonomy algebra bundle, the elements of which are special vector fields on the fibres of the given bundle.
Received: May 16, 2006; Revised: July 30, 2006; Accepted: August 2, 2006 相似文献
9.
Kang Hai TAN Xiao Ping YANG 《数学学报(英文版)》2006,22(3):701-710
In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli Garofalo-Nhieu and Pauls who recently introduced sub- Riemannian minimal surfaces in Carnot groups. This will be done by introducing a natural nonholonomic connection which is the restriction (projection) of the natural Riemannian connection on the horizontal bundle. For this nonholonomic connection and (intrinsic) regular hypersurfaces we introduce the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out that the horizontal mean curvature is the trace of the horizontal shape operator. 相似文献
10.
We study the projection p: Md ? Bd{\pi : \mathcal{M}_d \rightarrow \mathcal{B}_d} which sends an affine conjugacy class of polynomial
f : \mathbbC ? \mathbbC{f : \mathbb{C} \rightarrow \mathbb{C}} to the holomorphic conjugacy class of the restriction of f to its basin of infinity. When Bd{\mathcal{B}_d} is equipped with a dynamically natural Gromov–Hausdorff topology, the map π becomes continuous and a homeomorphism on the shift locus. Our main result is that all fibers of π are connected. Consequently, quasiconformal and topological basin-of-infinity conjugacy classes are also connected. The key
ingredient in the proof is an analysis of model surfaces and model maps, branched covers between translation surfaces which
model the local behavior of a polynomial. 相似文献
11.
We study the limiting behavior of the K?hler–Ricci flow on
\mathbbP(O\mathbbPn ?O\mathbbPn(-1)?(m+1)){{\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus(m+1)})}} for m, n ≥ 1, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses
to
\mathbbPn{{\mathbb{P}^n}} or contracts a subvariety of codimension m + 1 in the Gromov–Hausdorff sense. We also show that the K?hler–Ricci flow resolves a certain type of cone singularities
in the Gromov–Hausdorff sense. 相似文献
12.
V. V. Fedorchuk 《Journal of Mathematical Sciences》1996,80(5):2118-2129
For bicompact X, the topology of uniform convergence on C(X) coincides with the Vietoris topology on the function graphs.
The Vietoris topology on C(X) in turn is generated by the Hausdorff uniformity which is complete only for finite X. The supplement
CH(X) of the space C(X) with respect to the Hausdorff uniformity consists of lower semi-continuous multivalued mappings Φ :
X → ℝ with compact fibers. The paper studies the spaces CH(X) in the spaces of multivalued mappings. It is proved that CH(X) is a Q-manifold provided that X is a Peano continuum. Several results on the metrizability on bicompacta having the form
CH(X, I) are also obtained. Bibliography: 27 titles.
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 18, pp. 213–235, 1995. 相似文献
13.
We compute local Gromov–Witten invariants of cubic surfaces at all genera. We use a deformation a of cubic surface to a nef
toric surface and the deformation invariance of Gromov–Witten invariants. 相似文献
14.
Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold 总被引:1,自引:0,他引:1
Marian Ioan Munteanu 《Mediterranean Journal of Mathematics》2008,5(1):43-59
In this paper we study a Riemannian metric on the tangent bundle T(M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger–Gromoll metric and a compatible almost complex structure which confers a structure
of locally conformal almost K?hlerian manifold to T(M) together with the metric. This is the natural generalization of the well known almost K?hlerian structure on T(M). We found conditions under which T(M) is almost K?hlerian, locally conformal K?hlerian or K?hlerian or when T(M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find
an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from
T(M). Moreover, we found that this map preserves also the natural contact structures obtained from the almost Hermitian ambient
structures on the unit tangent bundle and the tangent sphere bundle, respectively.
This work was also partially supported by Grant CEEX 5883/2006–2008, ANCS, Romania. 相似文献
15.
We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental
group of the cotangent bundle, and of the Maslov class and second Stiefel–Whitney class of the Lagrangian submanifold) we
prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions
on their topology. An essentially equivalent result was recently proved independently by Nadler [16], using a different approach. 相似文献
16.
Y.-P. Lee 《Inventiones Mathematicae》2001,145(1):121-149
The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov–Witten invariants of X and Gromov–Witten invariants of complete intersections Y in X is established.
Oblatum 21-IV-2000 & 11-I-2001?Published online: 2 April 2001 相似文献
17.
One of our two main results exhibits for a vector bundle over a compact Hausdorff spaceX an interplay between its span, its possible splittings, and the Lyusternik-Shnirel’man category ofX. The other main result, also on vector bundles and the Lyusternik-Shnirel’man category, enables us to derive certain inequalities
connecting the immersion codimension, the stable span, and the Lyusternik-Shnirel’man category of a smooth closed manifold
which is not stably parallelizable. Our results are applicable in various situations of general interest. 相似文献
18.
Chikako Mese 《manuscripta mathematica》1999,100(3):375-389
In this paper, we study the structure of locally compact metric spaces of Hausdorff dimension 2. If such a space has non-positive
curvautre and a local cone structure, then every simple closed curve bounds a conformal disk. On a surface (a topological
manifold of dimension 2), a distance function with non-positive curvature and whose metric topology is equivalent to the surface
topology gives a structure of a Riemann surface. The construction of conformal disks in these spaces uses minimal surface
theory; in particular, the solution of the Plateau Problem in metric spaces of non-positive curvature.
Received: 18 November 1997/ Revised versions: 15 January and 7 June 1999 相似文献
19.
We define a natural semi-definite metric on quasi-fuchsian space, derived from geodesic current length functions and Hausdorff
dimension, that extends the Weil–Petersson metric on Teichmüller space. We use this to describe a metric on Teichmüller space
obtained by taking the second derivative of Hausdorff dimension and show that this metric is bounded below by the Weil–Petersson
metric. We relate the change in Hausdorff dimension under bending along a measured lamination to the length in the Weil–Petersson
metric of the associated earthquake vector of the lamination.
Martin Bridgeman research supported in part by NSF grant DMS 0305634. Edward C. Taylor research supported in part by NSF grant
DMS 0305704. 相似文献
20.
We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points
Gromov–Witten invariants can always be interpreted as classical intersection numbers on auxiliary varieties. Our main combinatorial
tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincaré duality.
In particular, we compute the quantum cohomology of the two exceptional minuscule homogeneous varieties.
DOI: . 相似文献