[Paper Review] Lipschitz-Volume rigidity in Alexandrov geometry

Nan Li|arXiv (Cornell University)|Oct 25, 2011

Geometric Analysis and Curvature Flows7 references3 citations

TL;DR

This paper establishes a Lipschitz-Volume rigidity theorem in Alexandrov geometry: if a 1-Lipschitz map between $n$-dimensional Alexandrov spaces preserves volume, it must be a path isometry, and its restriction to the interior is an isometry. The key result characterizes the metric structure of the target space when the map is surjective, proving that gluing along boundaries must be isometric—thus confirming a conjecture of Petrunin on the necessity of isometric gluing for preserving Alexandrov curvature bounds.

ABSTRACT

We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\colon X=\amalg X_\ell o Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of $X$. We furthermore characterize the metric structure on $Y$ with respect to $X$ when $f$ is also onto. This implies the converse of Petrunin's Gluing Theorem: if a gluing of two Alexandrov spaces via a bijection between their boundaries produces an Alexandrov space, then the bijection must be an isometry.

Motivation & Objective

• To establish a rigidity theorem for 1-Lipschitz, volume-preserving maps between Alexandrov spaces.
• To characterize the metric structure of the target space when such a map is surjective.
• To resolve a conjecture by Petrunin on the necessity of isometric gluing for constructing Alexandrov spaces.
• To provide necessary conditions for volume-preserving gluing of Alexandrov spaces via metric structure analysis.
• To generalize volume rigidity results to singular spaces with boundary, extending known results for manifolds and Ricci limit spaces.

Proposed method

• Use of the disjoint union of Alexandrov spaces $X = igsqcup X_ u$ to model gluing structures.
• Application of the intrinsic metric and Hausdorff measure to define volume and path length.
• Leverage the fact that volume preservation and 1-Lipschitz condition imply length preservation along paths.
• Use of Perel’man’s stability theorem to analyze Gromov-Hausdorff limits of spaces with maximal volume.
• Apply the theory of metric space gluing and isometric involutions on boundary spheres to characterize limit spaces.
• Use of Smith theorem and Poincaré duality to deduce topological type (e.g., homotopy sphere) of self-glued spaces.

Experimental results

Research questions

• RQ1 Under what conditions does a 1-Lipschitz, volume-preserving map between Alexandrov spaces become a path isometry?
• RQ2 What metric structure must the target space have if such a map is also surjective?
• RQ3 Is isometric gluing of boundaries necessary for the resulting space to remain an Alexandrov space?
• RQ4 Can volume-maximal Alexandrov spaces be characterized as self-glued metric cones via isometric involutions?
• RQ5 What topological constraints arise when a space is constructed by gluing Alexandrov spaces along their boundaries with volume preservation?

Key findings

• A 1-Lipschitz, volume-preserving map $f: X = igsqcup X_ u o Y$ between $n$-dimensional Alexandrov spaces is a path isometry.
• The restriction of $f$ to the interior of $X$ is an isometry with respect to the intrinsic metrics.
• If $f$ is surjective, then the metric on $Y$ is induced by the gluing $x_1 hicksim x_2 ext{ iff } f(x_1) = f(x_2)$, and the gluing is isometric.
• The preimage of any point in $Y$ is finite, and points with multiple preimages lie entirely in the boundary of $X$.
• The set of points in $Y$ with exactly two preimages has Hausdorff dimension $n-1$, and the set with three or more preimages has dimension at most $n-2$.
• The converse of Petrunin’s Gluing Theorem holds: if gluing two Alexandrov spaces via a bijection between boundaries yields an Alexandrov space, then the bijection must be an isometry.