# Seminars & Colloquia Calendar

Nonlinear Analysis

## Total Curvature and the isoperimetric inequality in negatively curved manifolds

#### Joel Spruck, Johns Hopkins University

Location:  Hill 705
Date & time: Tuesday, 22 October 2019 at 1:40PM - 2:50PM

Abstract: We prove that the total positive Gauss-Kronecker curvature of any closed hypersurface embedded in a complete simply connected manifold of nonpositive curvature $$Mn,\,n\geq 2$$, is bounded below by the volume of the unit sphere in Euclidean space $$R^n$$.

This yields the optimal isoperimetric inequality for bounded regions of finite perimeter in $$M$$, and thus settles the Cartan-Hadamard conjecture. Our starting point is a comparison formula for total curvature of level sets in Riemannian manifolds.

This is joint work with Mohammad Ghomi.

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