The isoperimetric and Willmore problems

Abstract

We introduce some methods to study the isoperimetric problem in 3-dimensional Riemannian manifolds and we show that in the positive curva- ture case we can control the topology of the isoperimetric regions. We consider the case of the projective space, which was first solved by Ritorea nd Ros, and we apply it to the Willmore problem. The isoperimetric problem is a classical topic in geometry but at the same time many basic questions about it remain unsolved. In this paper we first introduce some of the methods used in the study of that problem, although we will not try to be exhaustive at all. We will explain some relatively flexible ideas, like symmetrization or stability, which can be adapted to a certain number of situations. As an example we will study the problem for radial metrics on the 3-sphere. Then we will show that in 3-manifolds with positive Ricci curvature the topol- ogy of the isoperimetric regions can be controled. In particular we will prove that, when the volume of the ambient space is large, any isoperimetric surface must be either an sphere or a torus. As consequence, we will solve the isoperimetric problem in the real projective space or, equivalently, the isoperimetric problem for antipodal invariant regions in the 3-sphere. That result was first obtained by Ritorea nd Ros (35), but here we will give a somewhat different proof. Finally, as application of the above results, we will solve the Willmore conjec- ture for tori in euclidean space which are symmetric with respect to a point. 2. The isoperimetric problem In this paper we will only consider the three dimensional case. Let M be a Riemaniann 3-dimensional manifold with or without boundary and volume V(M ) ∈ )0, ∞). Given a positive number v< V(M ), we want to study the compact surfaces Σ ⊂ M such that