Isoperimetric Inequalities in Riemannian Manifolds

Manuel Ritoré is professor of Mathematics at the University of Granada since 2007. His earlier research focused on geometric inequalities in Riemannian manifolds, specially on those of isoperimetric type. In this field he has obtained some results such as a classification of isoperimetric sets in the 3-dimensional real projective space; a classification of 3-dimensional double bubbles; existence of solutions of the Allen-Cahn equation near non-degenerate minimal surfaces; an alternative proof of the isoperimetric conjecture for 3-dimensional Cartan-Hadamard manifolds; optimal isoperimetric inequalities outside convex sets in the Euclidean space; and a characterization of isoperimetric regions of large volume in Riemannian cylinders, among others. Recently, he has become interested on geometric variational problems in spaces with less regularity, such as sub-Riemannian manifolds or more general metric measure spaces, where he has obtained a classification of isoperimetric sets inthe first Heisenberg group under regularity assumptions, and Brunn-Minkowski inequalities for metric measure spaces, among others.

• - 1. Introduction. - 2. Isoperimetric Inequalities in Surfaces. - 3. The Isoperimetric Profile of Compact Manifolds. - 4. The Isoperimetric Profile of Non-compact Manifolds. - 5. Symmetrization and Classical Results. - 6. Space Forms. - 7. The Isoperimetric Profile for Small and Large Volumes. - 8. Isoperimetric Comparison for Sectional Curvature. - 9. Relative Isoperimetric Inequalities.