Calculus of Variations and Geometric Measure Theory

M. Che - F. Galaz-Garcia - M. Kerin - J. Santos-Rodríguez

Isometric Rigidity of Metric Constructions with respect to Wasserstein Spaces

created by chemoguel on 30 Jan 2025

[BibTeX]

preprint

Inserted: 30 jan 2025

Year: 2024

ArXiv: 2410.14648 PDF

Abstract:

In this paper we study the isometric rigidity of certain classes of metric spaces with respect to the $p$-Wasserstein space. We prove that spaces that split a separable Hilbert space are not isometrically rigid with respect to $\mathbb{P}_2$. We then prove that infinite rays are isometrically rigid with respect to $\mathbb{P}_p$ for any $p\geq 1$, whereas taking infinite half-cylinders (i.e.\ product spaces of the form $X\times [0,\infty)$) over compact non-branching geodesic spaces preserves isometric rigidity with respect to $\mathbb{P}_p$, for $p>1$. Finally, we prove that spherical suspensions over compact spaces with diameter less than $\pi/2$ are isometrically rigid with respect to $\mathbb{P}_p$, for $p>1$.