The optimal isoperimetric inequality in Cartan-Hadamard's manifolds and the Aubin conjecture.
Based on the article “Total curvature and isoperimetric inequality in the Cartan-
Hadamard manifolds”, by Mohammad Ghomi and Joel Spruck, we study a
comparison formula for the total curvature of level sets in Riemannian manifolds. In
particular, for cases where a manifold has constant sectional curvature, or for geodesic
balls in a manifold with a sectional curvature bounded above by a negative real constant. With the generalized Kleiner method, this comparison formula was applied to
the isoperimetric problem in spaces of non-positive curvature in order to obtain an
equivalent version for the Aubin conjecture.