Mohammad Ghomi

Geometric inequalities in spaces of nonpositive curvature

The classical isoperimetic inequality states that among all regions with a given perimeter in Euclidean space balls enclose the most volume. Generalizing this fact to spaces of nonpositive curvature has been an outstanding problem in Riemannian geometry, which is known as the Cartan-Hadamard conjecture. We will discuss a number of related results, and methods developed in recent years to study this problem. Central to these investigations are variational techniques which lead naturally to study of total mean curvatures, or integrals of symmetric functions of the principle curvatures of hypersurfaces. These quantities, which are also known as quermassintegrals,  appear in Steiner’s formula, Brunn-Minkowski theory, and Alexandrov-Fenchel inequalities. We will describe a number of new inequalities for these integrals in nonpositively curved spaces, which are obtained via Reilly’s identities, Chern-type differential forms, and harmonic mean curvature flow. As applications we obtain several new isoperimetric inequalities, and Riemannian rigidity theorems. This is joint work with Joel Spruck.