Special Year 2018-19: Variational Methods in Geometry - Seminar

Abstract: I will go over some recent work that I have been involved in on surface geometry in complete locally homogeneous 3-manifolds X. In joint work with Mira, Perez and Ros, we have been able to finish a long term project related to the Hopf...

In this talk we will survey recent progress on the Beresticky-Caffarelli-Nirenberg Conjecture in Space Forms; that is, let $\Omega$ be an open connected domain of a complete connected Riemannian manifold ($M,g$) and consider the OEP given by
\begin...

Preliminary I will expose a technique developed with T. Rivi\`{e}re to prove energy identities (weak compactness) for sequences of solutions of any conformally invariant problem of second order in dimension 2, see [1]. Then after introducing some...

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...

We'll describe a joint project with X. Zhou in which we use min-max techniques to prove existence of closed hypersurfaces with prescribed mean curvature in closed Riemannian manifolds. Our min-max theory handles the case of nonzero constant mean...

Following a program proposed by Gromov, we study metric singularities of positive scalar curvature of codimension two and three. In addition, we describe a comparison theorem for positive scalar curvature that is captured by polyhedra. Part of this...

In the 80s Pitts-Rubinstein conjectured that certain kinds of Heegaard surfaces in three-manifolds can be isotoped to index 1 minimal surfaces. I’ll describe in detail a proof of their conjecture and some applications. This is joint work with...

We consider the classical problem of prescribing the scalar curvature of a manifold via conformal deformation of the metric, dating back to works by Kazdan and Warner. This problem is mainly understood in low dimensions, where blow-ups of solutions...