Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 131, 29 pages      arXiv:2005.07435      https://doi.org/10.3842/SIGMA.2020.131
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

Annegret Burtscher a, Christian Ketterer b, Robert J. McCann b and Eric Woolgar c
a) Department of Mathematics, IMAPP, Radboud University, PO Box 9010, Postvak 59, 6500 GL Nijmegen, The Netherlands
b) Department of Mathematics, University of Toronto, 40 St George St, Toronto Ontario, Canada M5S 2E4
c) Department of Mathematical and Statistical Sciences and Theoretical Physics Institute, University of Alberta, Edmonton AB, Canada T6G 2G1

Received June 03, 2020, in final form November 21, 2020; Published online December 10, 2020

Abstract
Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

Key words: curvature-dimension condition; synthetic mean curvature; optimal transport; comparison geometry; diameter bounds; singularity theorems; inscribed radius; inradius bounds; rigidity; measure contraction property.

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