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Mathematics Colloquium
Total Curvature and the isoperimetric inequality: proof of the Cartan-Hadamard conjecture
Joel Spruck (Johns Hopkins University)

Thursday, January 23, 2020
3:30pm – 4:30pm

Storrs Campus
MONT 214

The classical isoperimetric inequality in Euclidean space is equiva- lent to the statement that spheres provide unique enclosures of least

perimeter for any given volume. In this talk we sketch some of the ideas

of the proof that this inequality also holds in complete simply con- nected spaces of nonpositive curvature, known as Cartan-Hadamard

manifolds. Immediate applications include sharp extensions of the Sobolev and Faber-Krahn inequalities to Cartan-Hadamard manifolds.

The essential step in the proof is to show that the total positive Gauss- Kronecker curvature of any closed hypersurface embedded in a Cartan- Hadamard manifold Mn

, n ≥ 2, is bounded below by the volume of the

unit sphere in Euclidean space Rn

. Our starting point is a comparison formula for the total curvature of level sets in Riemannian manifolds. This is joint work with Mohammad Ghomi.


Kyu-Hwan Lee

Mathematics Colloquium (primary), UConn Master Calendar

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