Analysis and Mathematical Physics

In the '80s, Yau conjectured that every closed Riemannian 3-manifold contains infinitely many smooth minimal hypersurfaces. In this talk, I will present a Yau-type existence result for nonlocal minimal surfaces and discuss how one can recover the classical Yau's conjecture from it. The main tools are min-max methods for the fractional perimeter and a compactness/regularity theory for the associated critical points, which gives smoothness in low dimensions and a small singular set in higher dimensions. Time permitting, I will discuss a broader program about extending these ideas to higher codimension.