# Lipschitz Lectures

## Closed minimal hypersurfaces in Riemannian manifolds

## Camillo De Lellis

University: Universität Zürich

**Local Organizer:** László Székelyhidi

**Date:** 15 May - 29 May 2009

**Location:** Wegelerstraße 10, University of Bonn, Germany

**Schedule:**

Friday, May 15, 10:15 – 12:00, Zeichensaal

Monday, May 18, 12:15 – 14:00, Kleiner Hörsaal

Wednesday, May 20, 10:15 – 12:00, Kleiner Hörsaal

Friday, May 22, 10:15 – 12:00, Zeichensaal

Monday, May 25, 12:15 – 14:00, Kleiner Hörsaal

Wednesday, May 27, 10:15 – 12:00, Kleiner Hörsaal

### Abstract

In 1917 Birkhoff proved the existence of a nontrivial closed geodesic in any Riemannian *2*-dimensional manifold diffeomorphic to the sphere. This result was later improved in a famous work by Ljusternik and Shnirelman to the existence of *three* distinct closed geodesics. The interest of Birkhoff's theorem lies on the fact that it cannot be achieved by standard "minimization arguments": since any closed curve in the sphere is contractible, minimizers of the length are necessarily trivial. In fact, much more can be proved: if the curvature of the metric is positive, there is no stable closed geodesic.

It is natural to ask whether the method of Birkhoff, a "min-max argument", can be extended to higher dimensions to produce nontrivial minimal hypersurfaces in general Riemannian manifolds. The answer to this question is positive and has quite important applications to other problems in geometry and topology. However, the difficulties in generalizing Birkhoff's argument are many, since the area functional does not enjoy good functional analytic properties. Indeed, the first proof of the existence of an (immersed) minimal *2*-sphere in any Riemannian manifold diffeomorphic to the *3*-sphere appeared only in 1981 in a celebrated paper by Sacks and Uhlenbeck.

Much research in the area has been triggered by a monograph of Pitts, who proved the existence of smooth embedded closed minimal hypersurface in any closed compact Riemannian manifold of dimension at most 6. Pitts' monograph appeared in 1981 and his result was then generalized to all dimensions by Schoen and Simon.

In a recent work with Dominik Tasnady we have introduced an approach similar to that of Pitts which however shortens the proof dramatically. The aim of this course is to give an account of this proof and of several tools of geometric measure theory which are used in it. I will therefore give short introductions to topics like the theory of Caccioppoli sets, the theory of varifolds and the curvature estimates for stable minimal hypersurfaces (due to works of Schoen, Simon and Yau).