Variational Methods in Geometry

I will outline the proof of density of minimal hypersurfaces (Irie-Marques-Neves) and equidistribution of minimal hypersurfaces (Marques-Neves-Song).

I will first survey some recent progress on global problems related to scalar curvature and area/volume, focusing in particular on scale breaking phenomena in such problems. I will then discuss the role of the Hawking mass in the resolution of this...

We will discuss the bubbling and neck analysis for degenerating sequences of minimal hypersurfaces which, in particular, lead to qualitative relationships between the variational, topological and geometric properties of these objects. Our aim is to...

Back in the 70s, Gromov started to study the relationship between the Lipschitz constant of a map (also called the dilation) and its topology. The Lipschitz constant describes the local geometric features of the map, and the problem is to...

The (Simon-Smith) min-max width of a Riemannian three dimensional sphere is a geometric invariant that measures the tightest way, in terms of area, of sweeping out the three-sphere by two-spheres. In this talk, we will explore the properties of this...

In the 60's Almgren initiated a program for developing Morse theory on the space of flat cycles. I will discuss some simplifications, generalizations and quantitative versions of Almgren's results about the topology of the space of flat cycles and...

We will present the project of using the Willmore elastic energy as a quasi-Morse function to explore the topology of immersions of the 2-sphere into Euclidean spaces and explain how this relates to the classical theory of complete minimal surfaces...

A translator for mean curvature flow is a hypersurface $M$ with the property that translation is a mean curvature flow. That is, if the translation is $t\rightarrow M+t\vec{v}$, then the normal component of the velocity vector $\vec{v}$ is equal to...

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 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...