共查询到20条相似文献,搜索用时 15 毫秒
1.
V. Kapovitch 《Geometric And Functional Analysis》2002,12(1):121-137
We prove that iterated spaces of directions of a limit of a noncollapsing sequence of manifolds with lower curvature bound
are topologically spheres. As an application we show that for any finite dimensional Alexandrov space X
n
with there exists an Alexandrov space Y homeomorphic to X which cannot be obtained as such a limit.
Submitted: December 2000, Revised: March 2001. 相似文献
2.
The classical Jung theorem gives an optimal upper estimate for the radius of a bounded subset of R
n in terms of its diameter and the dimension. In this note we present an analogue of this result for metric spaces of curvature bounded above in the sense of Alexandrov. 相似文献
3.
Longzhi Lin 《Journal of Geometric Analysis》2011,21(2):429-454
In this paper, we show a local energy convexity of W
1,2 maps into CAT(K) spaces. This energy convexity allows us to extend Colding and Minicozzi’s width-sweepout construction to produce closed
geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the
Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting. 相似文献
4.
By an ℱK-convex function on a length metric space, we mean one that satisfies f
n
≥ −Kf on all unitspeed geodesics. We show that natural ℱK-convex (-concave) functions occur in abundance on metric spaces of curvature bounded above (below) by K in the sense of Alexandrov. We prove Lipschitz extension and approximation theorems for ℱK-convex functions on CAT(K) spaces.
Received: 10 May 2002
Mathematics Subject Classification (2000): 53C70, 52A41 相似文献
5.
V. N. Berestovskii 《Journal of Mathematical Sciences》1999,94(2):1145-1146
The paper is devoted to discovering relations between inner metrics of manifolds with sectional curvature bounded from above
(from below) by some constant and topological (simplicial, piecewise-linear, or smooth) structures on such manifolds. Examples
of Alexandrov spaces with one-sided bounded curvature having exotic properties are given. Bibliography: 6 titles.
Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 17–19. Original 相似文献
6.
Stephanie Halbeisen 《manuscripta mathematica》2000,103(2):169-182
The tangent cones of an inner metric Alexandrov space with finite Hausdorff dimension and a lower curvature bound are always inner metric spaces with nonnegative curvature. In this paper we construct an infinite-dimensional inner metric Alexandrov
space of nonnegative curvature which has in one point a tangent cone whose metric is not an inner metric.
Received: 20 October 1999 / Revised version: 8 May 2000 相似文献
7.
Emanuel Milman 《Comptes Rendus Mathematique》2012,350(19-20):897-902
We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which are extremal for the isoperimetric problem. In particular, we recover the Gromov–Lévy and Bakry–Ledoux isoperimetric inequalities, which state that whenever the curvature is strictly positively bounded from below, these model spaces are the n-sphere and Gauss space, corresponding to generalized dimension being n and ∞, respectively. In all other cases, which seem new even for the classical Riemannian-volume measure, it turns out that there is no single model space to compare to, and that a simultaneous comparison to a natural one parameter family of model spaces is required, nevertheless yielding a sharp result. 相似文献
8.
S.-i. Ohta 《Mathematische Zeitschrift》2003,244(1):47-65
We prove that a totally geodesic map between a Riemannian manifold and a metric space can be represented as the composite
of a totally geodesic map from a Riemannian manifold to a Finslerian manifold and a locally isometric embedding between metric
spaces. As a corollary, we obtain the homotheticity of a totally geodesic map from an irreducible Riemannian manifold to an
Alexandrov space of curvature bounded above. This is a generalization of the case between Riemannian manifolds.
Mathematics Subject Classification (2000): 53C20, 53C22, 53C24
Received: 14 March 2002; in final form: 6 May 2002 / / Published online: 24 February 2003 相似文献
9.
In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space (X, dX) with curvature bounded above by a constant κ (κ ? 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.
相似文献10.
Ayato Mitsuishi 《Geometriae Dedicata》2010,144(1):101-114
We prove a splitting theorem for Alexandrov space of nonnegative curvature without properness assumption. As a corollary,
we obtain a maximal radius theorem for Alexandrov spaces of curvature bounded from below by 1 without properness assumption.
Also, we provide new examples of infinite dimensional Alexandrov spaces of nonnegative curvature. 相似文献
11.
In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L
n
+1 as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike
hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L
n
+1 which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates
for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in
L
n
+1 which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and
of a generalization of it.
Received: 5 July 1999 相似文献
12.
In Alexandrov spaces of curvature bounded either above (CBA) or below (CBB), we obtain extrinsic curvature bounds on subspaces
associated with semiconcave functions. These subspaces play the role in singular geometry of submanifolds in Riemannian geometry,
and arise naturally in many different places. For CBA spaces, we obtain new intrinsic curvature bounds on subspaces. For CBB
spaces whose boundary is extrinsically curved, we strengthen Perelman’s concavity theorem for distance from the boundary,
deriving corollaries on sharp diameter bounds, contractibility, and rigidity. 相似文献
13.
We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov
spaces in dimensions 3 and 4. As a corollary we obtain the classification of closed, n-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric T
n−1 action. In contrast to the one- and two-dimensional cases, where it is known that an Alexandrov space is a topological manifold,
in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension
of
\mathbbRP2 \mathbb{R}{P^2} , which is not a manifold. 相似文献
14.
A Gauss Equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K. That is, a subspace of extrinsic
curvature ⩽ A, defined by a cubic inequality on the difference of arc and chord, has intrinsic curvature ⩽ K+A2. Sharp bounds on injectivity radii of subspaces, new even in the Riemannian case, are derived. 相似文献
15.
Jürgen Jost 《Calculus of Variations and Partial Differential Equations》1994,2(2):173-204
We show the existence of harmonic mappings with values in possibly singular and not necessarily locally compact complete metric length spaces of nonpositive curvature in the sense of Alexandrov. As a technical tool, we show that any bounded sequence in such a space has a subsequence whose mean values converge. We also give a general definition of harmonic maps between metric spaces based on mean value properties and-convergence. 相似文献
16.
Emily Proctor 《Annals of Global Analysis and Geometry》2012,41(1):47-59
We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only
isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to
the orbifold category of a similar result for manifolds proven by Grove, Petersen, and Wu. It follows that any Laplace isospectral
collection of orbifolds with sectional curvature uniformly bounded below and having only isolated singular points also contains
only finitely many orbifold homeomorphism types. The main steps of the argument are to show that any sequence from the collection
has subsequence that converges to an orbifold, and then to show that the homeomorphism between the underlying spaces of the
limit orbifold and an orbifold from the subsequence that is guaranteed by Perelman’s stability theorem must preserve orbifold
structure. 相似文献
17.
In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study
the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short).
Requiring that, for each time t ≥ 0, the evolving hypersurface M
t
meets such tgh orthogonally, we prove that: (a) the flow exists while M
t
does not touch the axis of rotation; (b) throughout the time interval of existence, (b1) the generating curve of M
t
remains a graph, and (b2) the averaged mean curvature is double side bounded by positive constants; (c) the singularity set
(if non-empty) is finite and lies on the axis; (d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature. 相似文献
18.
In this paper the comparison result for the heat kernel on Riemannian manifolds with lower Ricci curvature bound by Cheeger and Yau (1981) is extended to locally compact path metric spaces (X,d) with lower curvature bound in the sense of Alexandrov and with sufficiently fast asymptotic decay of the volume of small geodesic balls. As corollaries we recover Varadhan's short time asymptotic formula for the heat kernel (1967) and Cheng's eigenvalue comparison theorem (1975). Finally, we derive an integral inequality for the distance process of a Brownian Motion on (X,d) resembling earlier results in the smooth setting by Debiard, Geavau and Mazet (1975). 相似文献
19.
Yusuke Higuchi 《Journal of Graph Theory》2001,38(4):220-229
Regarding an infinite planar graph G as a discrete analogue of a noncompact simply connected Riemannian surface, we introduce the combinatorial curvature of G corresponding to the sectional curvature of a manifold. We show this curvature has the property that its negative values are bounded above by a universal negative constant. We also prove that G is hyperbolic if its curvature is negative. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 220–229, 2001 相似文献
20.
Karim Adiprasito 《Geometriae Dedicata》2012,159(1):267-275
Solving a long-standing open question of Zamfirescu, we will show that typical convex surfaces contain points of infinite curvature in all tangent directions. To prove this, we use an easy curvature definition imitating the idea of Alexandrov spaces of bounded curvature, and show continuity properties for this notion. 相似文献