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 |
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.