Annegret Burtscher, Christian Ketterer, Robert J. McCann, Eric Woolgar, “Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary”, SIGMA, 2020, том 16,131, 29 стр.

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

Annegret Burtscher^a, Christian Ketterer^b, Robert J. McCann^b, 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

Аннотация: 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.

Ключевые слова: curvature-dimension condition, synthetic mean curvature, optimal transport, comparison geometry, diameter bounds, singularity theorems, inscribed radius, inradius bounds, rigidity, measure contraction property.

MSC: 51K10, 53C21, 30L99, 83C75

Поступила: 3 июня 2020 г.; в окончательном варианте 21 ноября 2020 г.; опубликована 10 декабря 2020 г.

Язык публикации: английский

DOI: 10.3842/SIGMA.2020.131