The 22nd Takagi Lectures
November 17 (Sat), 15:00--16:00
November 18 (Sun), 11:30--12:30
Graduate School of Mathematical Sciences
The University of Tokyo, Tokyo, Japan


Abundance of minimal surfaces
Fernando Codá Marques
(Princeton University)


Abstract

These lectures concern the existence theory of closed minimal hypersurfaces in closed Riemannian manifolds. These hypersurfaces are critical points for the area functional, and hence their study can be seen as a high-dimensional generalization of the classical theory of closed geodesics (Birkhoff, Morse, Lusternik, Schnirelmann, ...). The best result until very recently, due to Almgren' 65 and Pitts' 81, was the existence of at least one closed minimal hypersurface in every closed Riemannian manifold.

I will discuss the methods I have developed with Andre Neves, for the past few years, to approach this problem through the variational point of view. These ideas have culminated with a series of dramatic developments in the field and the discovery that minimal hypersurfaces in fact abound.