DG-MP-PDE Seminar: Martin Man-chu Li

Abstract

For any smooth compact Riemannian 3-manifold with boundary, we prove that there always exists a smooth, embedded minimal surface with (possibly empty) free boundary. We also obtain a priori upper bound on the genus of such minimal surfaces in terms of the topology of the ambient compact 3-manifold. An interesting note is that no convexity assumption on the boundary is required. In this talk,  we will describe the min-max construction for the free boundary problem, and then we will sketch a proof of the existence part of the theory.