In Search of Minimal Hypersurfaces

Abstract

We study minimal hypersurfaces from the point of view of min-max theory. We present a proof of Yau's conjecture for the abundance of minimal surfaces, which builds on previous works by F. C. Marques and A. Neves, and extend it to some non-compact ambient manifolds. We show a generic equidistribution result for minimal hypersurfaces (joint with F. C. Marques and A. Neves). Then we give a proof of a conjecture by H. J. Rubinstein on realizing strongly irreducible Heegaard splittings of $3$-manifolds by minimal surfaces (joint with D. Ketover and Y. Liokumovich). Other results related to minimal surfaces are explained.