Geometry

A key problem in geometric measure theory is to understand singularities in minimal surfaces when multiplicity occurs in the tangent cone. For questions concerning partial regularity, the primary situation to understand is that of a branch point, namely a (non-immersed) singular point where one…

ALG gravitational instantons are complete hyper-K”ahler surfaces asymptotic to a twisted product of the complex plane and an elliptic curve. Following the classical work of Tian-Yau and Hein, etc., on Monge-Ampere methods for Ricci flat K”ahler metrics on quasi-projective varieties, we provide a…

 General relativists have long made their peace with “normal” black holes, even highly spinning ones. Extremal black holes, on the other hand, namely those having the maximum spin or charge consistent with being a black hole, remain a source of extreme uneasyness. I present a series of…

 The Riemannian Penrose inequality is a fundamental result in mathematical relativity. It has been a long-standing conjecture of G. Huisken that an analogous result should hold in the context of extrinsic geometry. In this talk, I will present recent joint work with M. Eichmair that…

A central problem in differential geometry is understanding how the geometry of a boundary determines the geometry of its interior. Gromov's fill-in problem suggests that when a closed Riemannian manifold is filled with a region of large curvature, the extrinsic curvature of the boundary must be…

We will illustrate by examples the role of canonical families in solving minmax problems. We will then propose a desingularisation of Marques-Neves-Ros canonical family leading to a new proof of the Willmore conjecture in dimension 3 and opening the way for considering the Willmore Problem in…

Hyperbolic reflection groups appear in various fields of mathematics such as algebraic geometry, discrete subgroups of Lie groups, geometric group theory, geometric topology, and number theory. Cofinite and cocompact hyperbolic reflection groups have the following feature: their fundamental…

The study of singularities of minimal submanifolds has a long history, with isolated singularities being the best understood case. The next simplest case is that of minimal submanifolds with families with singularities locally modeled on the product of an isolated conical singularity and a…

The Vafa Witten equations on 4-manifolds are the variational equations of a functional that generalizes one of the Chern-Simons functionals for SL(2;C) connections on 3-manifolds (and it reduces to that on products of a 3-manifold with the circle).  Being that the moduli space of solutions…

The existence problem for minimal hypersurfaces (or more generally prescribed mean curvature hypersurfaces) in complete noncompact manifolds is fundamental, but little is known. Min-max provides a powerful framework for existence in closed manifolds, but relies critically on compactness. I…