[Paper Review] Self-contracted curves in CAT(0)-spaces and their rectifiability

Shin‐ichi Ohta|arXiv (Cornell University)|Nov 25, 2017

Geometric Analysis and Curvature Flows19 references3 citations

TL;DR

This paper establishes the rectifiability of bounded self-contracted curves in $\mathrm{CAT}(0)$-spaces under uniform local structural conditions, generalizing results from Euclidean and Hadamard manifolds. It introduces three geometric conditions on the space's local structure—related to angles, volume growth, and directionality—that ensure bounded self-contracted curves have finite length, and verifies these in trees, books, and $\mathrm{CAT}(0)$-simplicial complexes, even with non-uniform dimensions.

ABSTRACT

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 self-contracted 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.

Motivation & Objective

• To extend the rectifiability of bounded self-contracted curves from Euclidean and Hadamard manifolds to general $\mathrm{CAT}(0)$-spaces.
• To identify uniform local geometric conditions on $\mathrm{CAT}(0)$-spaces that guarantee rectifiability of bounded self-contracted curves.
• To verify these conditions in concrete examples such as trees, books, and $\mathrm{CAT}(0)$-simplicial complexes.
• To demonstrate that the rectifiability result holds even in spaces with non-uniform local dimensions.
• To provide a quantitative refinement of the rectifiability argument in Hadamard manifolds using comparison geometry.

Proposed method

• Adapts and refines the proof strategy from Daniilidis et al. in Hadamard manifolds using comparison theorems to derive quantitative estimates on angle and distance decay.
• Introduces three uniform local conditions (I, II, III) on $\mathrm{CAT}(0)$-spaces: uniform bounds on angle distortion, volume growth in tangent cones, and directionality of geodesics.
• Applies these conditions to prove rectifiability in $\mathrm{CAT}(0)$-spaces with non-uniform local dimensions, such as a union of $\mathbb{R}^2$ and $[0,\infty)$ identified at a point.
• Uses volume estimates in tangent cones and bi-Lipschitz control of projections to verify the conditions in simplicial complexes.
• Employs comparison geometry in $\mathrm{CAT}(0)$-spaces to bound the length of self-contracted curves via angle estimates and metric distortion.
• Applies the theory to $\mathrm{CAT}(0)$-simplicial complexes by decomposing them into bounded simplexes and verifying the structural conditions (1), (3) in Theorem 6.4.

Experimental results

Research questions

• RQ1 Under what uniform geometric conditions on the local structure of a $\mathrm{CAT}(0)$-space are bounded self-contracted curves rectifiable?
• RQ2 Can the rectifiability of self-contracted curves in Hadamard manifolds be generalized to more general $\mathrm{CAT}(0)$-spaces using comparison geometry?
• RQ3 Do $\mathrm{CAT}(0)$-simplicial complexes with non-uniform local dimensions satisfy the conditions for rectifiability of self-contracted curves?
• RQ4 Is the self-contractedness property sufficient for rectifiability in $\mathrm{CAT}(0)$-spaces beyond finite-dimensional or smooth settings?
• RQ5 Can the rectifiability result be extended to $\mathrm{CAT}(1)$-spaces, particularly geodesically complete ones?

Key findings

• Bounded self-contracted curves in $\mathrm{CAT}(0)$-spaces satisfying the three uniform local conditions (I, II, III) have finite length.
• The rectifiability result holds in trees of bounded degree, books with finitely many sheets, and $\mathrm{CAT}(0)$-simplicial complexes under mild hypotheses.
• The conditions are verified in $\mathrm{CAT}(0)$-simplicial complexes by decomposing them into bounded simplexes and using volume and angle estimates in tangent cones.
• The method allows for spaces with non-uniform local dimensions, such as a space formed by gluing $\mathbb{R}^2$ and $[0,\infty)$ at the origin.
• The proof provides a quantitative refinement of the rectifiability argument in Hadamard manifolds via comparison geometry and angle estimates.
• The conditions are satisfied in geodesically complete $\mathrm{CAT}(0)$-spaces and can be extended to spaces isometrically embeddable into such spaces.