Total Curvature and the Isoperimetric Inequality: Proof of the Cartan-Hadamard Conjecture

Department of Mathematics Colloquium

Meeting Details

For more information about this meeting, contact Kristin Berrigan, Stephanie Geyer, Sergei Tabachnikov, Dmitri Burago, Alberto Bressan, Nigel Higson.

Speaker: Mohammad Ghomi, Georgia Institute of Technology

Abstract: The classical isoperimetric inequality states that in Euclidean space spheres provide unique enclosures of least perimeter for any given volume. In this talk we show that this inequality also holds in spaces of nonpositive curvature, called Cartan-Hadamard manifolds, as conjectured by Aubin, Gromov, Burago, and Zalgaller in 1970s and 80s. The proof is based on a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for the smooth approximation of the signed distance function, via inf-convolution and Reilly type formulas among other techniques. Immediate applications include sharp extensions of Sobolev and Faber-Krahn inequalities to spaces of nonpositive curvature. This is joint work with Joel Spruck.

Room Reservation Information

Room Number: 114 McAllister

Date: 11/07/2019

Time: 3:30pm - 4:30pm