Metric Spaces of Non-Positive Curvature

Bridson, Martin R.; Haefliger, André

doi:10.1007/978-3-662-12494-9

Cited by 3,405 publications

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“…De nition 2.2 (CAT(κ)-inequality, see [5]). Let (Y , d Y ) be a metric space and a geodesic triangle in Y with perimeter strictly less than Rκ.…”

Section: ) Where U(t) = T E T (U ) V(t) = T E T (V ) With U V ∈ D(e)mentioning

confidence: 99%

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Resolvent Flows for Convex Functionals and p-Harmonic Maps

Kuwae

2015

Analysis and Geometry in Metric Spaces

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Abstract:We prove the unique existence of the (non-linear) resolvent associated to a coercive proper lower semicontinuous function satisfying a weak notion of p-uniform λ-convexity on a complete metric space, and establish the existence of the minimizer of such functions as the large time limit of the resolvents, which generalizing pioneering work by Jost for convex functionals on complete CAT( )-spaces. The results can be applied to L p -Wasserstein space over complete p-uniformly convex spaces. As an application, we solve an initial boundary value problem for p-harmonic maps into CAT( )-spaces in terms of Cheeger type p-Sobolev spaces.

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“…De nition 2.2 (CAT(κ)-inequality, see [5]). Let (Y , d Y ) be a metric space and a geodesic triangle in Y with perimeter strictly less than Rκ.…”

Section: ) Where U(t) = T E T (U ) V(t) = T E T (V ) With U V ∈ D(e)mentioning

confidence: 99%

“…De nition 2.3 (CAT(κ)-space, see [5] The following notion of p-uniformly convex space is proposed by Naor-Silberman [23] in the framework of geodesic spaces. It has been formulated in terms of the modulus of convexity for Banach spaces.…”

Section: ) Where U(t) = T E T (U ) V(t) = T E T (V ) With U V ∈ D(e)mentioning

confidence: 99%

Resolvent Flows for Convex Functionals and p-Harmonic Maps

Kuwae

2015

Analysis and Geometry in Metric Spaces

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“…Then y = (y i ) is a well-de ned point of X, and lim k→∞ x k = y. Finally, it is well known that a complete space that is the ultralimit of proper geodesic spaces is also proper and geodesic (see either [2] or [3]). One can also consult the previous two references for the de nition of an ultralimit, but all that is important here is that being a pro-Euclidean space is stronger than being an ultralimit of polyhedra.…”

Section: Euclidean Polyhedra and Pro-euclidean Spacesmentioning

confidence: 99%

“…It is well known (see [2] and [3]) that a complete metric space which is an ultralimit of proper metric spaces is also proper. Then since a pro-Euclidean space is, in particular, an ultralimit of locally-nite polyhedra, what needs to be shown is that any pro-Euclidean space is complete.…”

Section: Introductionmentioning

confidence: 99%

Isometric Embeddings of Pro-Euclidean Spaces

Minemyer

2015

Analysis and Geometry in Metric Spaces

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Abstract:In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into E n if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a "nice" inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E n+ . Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C version of) the famous Nash isometric embedding theorem from [10].

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“…Corollary 4.5 gives us enough information to apply the following lemma, the statement of which is recorded from [4], and deduce sufficiently high powers of n independent pseudo-Anosov mapping classes freely generate a free group of rank n.…”

Section: Ping-ponging In Teichmüller Spacementioning

confidence: 99%