Hypersurfaces and Isoperimetric Inequalities in Cartan-Hadamard Manifolds

Contact for zoom details: Christian Kaiser (kaiser@mpim-bonn.mpg.de)

Complete, simply connected Riemannian manifolds with non-positive sectional curvature are known as Cartan-Hadamard manifolds.  These spaces are fundamental in global differential geometry and many of its interactions with analysis, topology and group theory.  There is a long-standing conjeture that the isoperimetric inequality from Euclidean space, relating the volume of a bounded domain to the area of its perimeter, also holds in Cartan-Hadamard manifolds.  The conjecture is currently open in all dimensions greater than four.  We will prove that a closely related inequality, known as the Banchoff-Pohl inequality, holds in Cartan-Hadamard manifolds of all dimensions.  We will also discuss the connections between this result and several other problems in geometry and analysis.