Variational Methods in Geometry

I will discuss a proof of the existence of infinitely many solutions for the singular Yamabe problem in spheres using bifurcation theory and the spectral theory of hyperbolic surfaces.

In recent works with N. Wickramasekera we develop a regularity and compactness theory for stable hypersurfaces (technically, integral varifolds) whose generalized mean curvature is prescribed by a (smooth enough) function on the ambient Riemannian...

I will present a proof with some substantial details of the Multiplicity One Conjecture in Min-max theory, raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal...

In this talk, we will be concerned with the existence of a third embedded minimal hypersurface spanning a closed submanifold B contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence...

I will present a proof with some substantial details of the Multiplicity One Conjecture in Min-max theory, raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal...

In this talk I want to discuss two related questions about the variational structure of the Yang-Mills functional in dimension four. The first is the question of 'gap' estimates; i.e., determining an energy threshold below which any solution must...

A decades-old application of the second variation formula proves that if the scalar curvature of a closed 3--manifold is bounded below by that of the product of the hyperbolic plane with the line, then every 2--sided stable minimal surface has...

Geodesic nets on Riemannian manifolds is a natural generalization of geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere.

In...

Anisotropic surface energies are a natural generalization of the perimeter functional that arise in models in crystallography and in scaling limits for certain probabilistic models on lattices. This talk focuses on two results concerning...

We review various recent results aimed at understanding bubbling into spheres for boundaries with almost constant mean curvature. These are based on joint works with Giulio Ciraolo (U Palermo), Matias Delgadino (Imperial College London), Brian...