Gluing construction for infinitely many minimal surfaces. Joint with D. Stern, we proved that every Riemannian 3-manifold with generic metric contains infinitely minimal surfaces with bounded area [link]. Our proof uses the doubling construction developed by Kapouleas-McGarth. In particular, this gives a new solution to Yau's conjecture (1982) in the generic metric setting.
Enumerative problems for minimal surfaces of prescribed genus. It was conjectured by B. White (1989) that every Riemannian 3-sphere contains at least 5 minimal tori. Joint with Y. Li, we confirmed this for metrics of positive Ricci curvature using min-max theory [link]. Recently, I also showed that every Ricci-positive 3-sphere contains a genus g minimal surface for every g [link]. See this page for an introduction. These results rely on studying the topology of the set of all (singular) surfaces embedded in S3.
Singularities in geometric flow. Together with A. Sun, we developed topological techniques to construct genus one singularities in mean curvature flow [link]. As an application, we found a genus one self-shrinker in ℝ3 with entropy less than the shrinking doughnut.
