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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.