Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces
Differential Geometry
2024-04-26 v3 Functional Analysis
Metric Geometry
Abstract
In this paper we consider metric fillings of convex bodies. We show that convex bodies are the unique minimal fillings of their boundary metrics among all integral current spaces. To this end, we also prove that convex bodies enjoy the Lipschitz-volume rigidity property within the category of integral current spaces, which is well known in the smooth category. As a further application of this result, we answer a question of Perales concerning the intrinsic flat convergence of minimizing sequences for the Plateau problem.
Keywords
Cite
@article{arxiv.2209.12545,
title = {Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces},
author = {Giuliano Basso and Paul Creutz and Elefterios Soultanis},
journal= {arXiv preprint arXiv:2209.12545},
year = {2024}
}
Comments
25 pages, 1 figure, v3: we have added Corollary 1.3 concerning the LV-rigidity of the sphere. In version 2 we had already added Theorem 1.1, a filling minimality result for convex bodies