Antoine Song

Princeton University

October 23, 2018

In the early 80’s, Yau conjectured that in any closed 3-manifold there should be infinitely many minimal surfaces. I will review previous contributions to the question and present a proof of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves. A key step is the construction by min-max theory of a sequence of closed minimal surfaces in a manifold N with non-empty stable boundary, and I will explain how to achieve this via the construction of a non-compact cylindrical manifold. Then, after proving that the area growth of this sequence of minimal surfaces is linear and that the leading factor is the area of the largest boundary component of N, we can end the proof by contradiction.