Isoperimetric inequality in spaces of nonpositive curvative

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