R202, Astronomy-Mathematics Building, NTU

(台灣大學天文數學館 202室)

Heintze-Karcher-Ros Inequalities and the Soap Bubble Theorems in Geometric Measure and Convex Geometry

Mario Santilli (University of Augsburg)

Abstract:

One of the most celebrated result in differential geometry asserts that a compact, embedded, smooth hypersursurface in

**R**^{n}^{+1} is a sphere, provided it has constant scalar mean curvature. This is the famous Alexandrov soap bubble theorem. It was later discovered by Ros that this result (as well as its generalization to constant higher-order mean curvatures) is a special case of an isoperimetric-type result, which asserts that if

is a bounded open set with smooth boundary and

has non-negative scalar mean curvature

(i.e.

is a mean convex set), then

,

with equality if and only if

is a sphere.

In this talk I will present several generalizations of this isoperimetric result to classes of mean convex sets arising from different areas of geometric analysis, where the smoothness assumption is no longer guaranteed. They include all convex bodies, the mean convex sets introduced by White in the regularity theory of the mean curvature flow, and the sets of finite perimeter whose reduced boundary is a varifold with bounded anisotropic distributional mean curvature. In particular, we partially answer a conjecture of Maggi, generalizing the Alexandrov soap bubble theorem to stationary points of the anisotropic isoperimetric problem for smooth elliptic integrands.

Part of the results are a joint work with S\l{}awomir Kolasinski (Warsaw) and Antonio De Rosa (Courant Institute).

Reference:

[1] Francesco Maggi. Critical and almost-critical points in isoperimetric problems. *Oberwolfach Rep*., 35:34--37, 2018.

[2] Antonio~De Rosa, Slawomir Kolasinski, and Mario Santilli. Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets, 2019. *arXiv e-prints*, page arXiv:1908.09795, Aug 2019.

[3] Mario Santilli. The heintze-karcher-ros inequality and the soap bubble theorem in geometric measure and convex geometry, 2019. *arXiv e-prints*, page arXiv:1908.05952, Aug 2019.

Note:

The presentation will be conducted using Cisco Webex meeting and the meeting coordinates maybe obtained by contacting one of the organisers of the seminar.