Geodesically complete spaces with an upper curvature bound

Lytchak, Alexander; Nagano, Koichi

doi:10.48550/arxiv.1804.05189

Cited by 5 publications

(16 citation statements)

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“…This is immediate from the fact that a pointed neighborhood of an interior point cannot be contractible. Hence the interior of a CAT(0) surface is GCBA in the sense of Lytchak and Nagano [LN18].…”

Section: Cat(0) Surfacesmentioning

confidence: 97%

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The structure of minimal surfaces in CAT(0) spaces

Stadler¹

2018

Preprint

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Section: Cat(0) Surfacesmentioning

confidence: 97%

“…Proof. By Theorem 12.1 in [LN18] it is enough to show that no measure is concentrated near the boundary.…”

Section: Cat(0) Surfacesmentioning

confidence: 99%

The structure of minimal surfaces in CAT(0) spaces

Stadler¹

2018

Preprint

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“…In section 3 we develop a structure theory for general RCD + CAT spaces where we adapt the DC-calculus of Lytchak-Nagano [LN18]. This might be of independent interest.…”

Section: Since Smooth Riemannian Manifolds Locally Have Curvature Bou...mentioning

confidence: 99%

Weakly Noncollapsed RCD Spaces with Upper Curvature Bounds

Kapovitch

Ketterer

2019

Analysis and Geometry in Metric Spaces

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We develop a structure theory for RCD spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further the set of regular points is a smooth manifold and is geodesically convex. Around regular points there are DC coordinates and the distance is induced by a continuous BV Riemannian metric.

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“…Although we do not have definite bounds in general, a compactness argument may verify (some of) these conditions under appropriate assumptions (such as the geodesic completeness, in other words, the infinite extendability of geodesics). We remark that local structures of CAT(0)-spaces are investigated in [Kl] as well as in a famous unpublished paper by Otsu-Tanoue ("The Riemannian structure of Alexandrov spaces with curvature bounded above") and Lytchak-Nagano's recent preprints [LN1,LN2].…”

Section: Examplesmentioning

confidence: 99%

Self-contracted curves in CAT(0)-spaces and their rectifiability

Ohta

2017

Preprint

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We investigate self-contracted curves, arising as (discrete or continuous time) gradient curves of quasi-convex functions, and their rectifiability (finiteness of the lengths) in Euclidean spaces, Hadamard manifolds and CAT(0)-spaces. In the Hadamard case, we give a quantitative refinement of the original proof of the rectifiability of bounded selfcontracted curves (in general Riemannian manifolds) by Daniilidis et al. Our argument leads us to a generalization to CAT(0)-spaces satisfying several uniform estimates on their local structures. Upon these conditions, we show the rectifiability of bounded self-contracted curves in trees, books and CAT(0)-simplicial complexes.

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Geodesically complete spaces with an upper curvature bound

Cited by 5 publications

References 0 publications

The structure of minimal surfaces in CAT(0) spaces

The structure of minimal surfaces in CAT(0) spaces

Weakly Noncollapsed RCD Spaces with Upper Curvature Bounds

Self-contracted curves in CAT(0)-spaces and their rectifiability

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