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.

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