Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces-Reference-Cited by-同舟云学术

Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces

Published:2023-11-01 Issue:0 Volume:0 Page:
ISSN:0075-4102
Container-title:Journal für die reine und angewandte Mathematik (Crelles Journal)
language:en
Short-container-title:

Author:

Basso Giuliano¹^ORCID,Creutz Paul¹,Soultanis Elefterios²^ORCID

Affiliation:

1. Max Planck Institute for Mathematics , Vivatsgasse 7, 53111 Bonn , Germany

2. Department of Mathematics and Statistics , University of Jyväskylä , Mattilanniemi (MaD), 40014 , Jyväskylä , Finland

Abstract

Abstract In this paper we consider metric fillings of boundaries 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 further applications of this result, we prove a variant of Lipschitz-volume rigidity for round spheres and answer a question of Perales concerning the intrinsic flat convergence of minimizing sequences for the Plateau problem.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/crelle-2023-0076/pdf

Reference49 articles.

1. B. Allen and R. Perales, Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below, preprint (2020), https://arxiv.org/abs/2006.13030.

2. B. Allen, R. Perales and C. Sormani, Volume above distance below, preprint (2022), https://arxiv.org/abs/2003.01172v3; to appear in J. Differential Geom.

3. J. C. Álvarez Paiva and A. C. Thompson, Volumes on normed and Finsler spaces, A sampler of Riemann–Finsler geometry, Math. Sci. Res. Inst. Publ. 50, Cambridge University, Cambridge (2004), 1–48.

4. L. Ambrosio and B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), no. 1, 1–80.

5. K. Ball, Ellipsoids of maximal volume in convex bodies, Geom. Dedicata 41 (1992), no. 2, 241–250.