# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Embeddedness and convexity for hypersurfaces in hyperbolic space

I will talk a proof of the conjecture of Alexander and Currier on the embeddedness of a nonnegatively curved hypersurfaces in hyperbolic space. I will also discuss some recent works on hypersurfaces with nonnegative Ricci curvature in hyperbolic space.

##### Stable CMC hypersurfaces

In a joint work with N. Wickramasekera (Cambridge) we develop a regularity and compactness theory for a class of codimension-1 integral n-varifolds with generalised mean curvature in L^{p}_{loc} for some p > n. Subject to suitable variational hypotheses on the regular part (namely stationarity and stability for the area functional with respect to variations that preserve the "enclosed volume") and two necessary structural assumptions, we show that the varifolds under consideration are "smooth" (and have constant mean curvature in the classical sense) away from a closed singular set of codimension 7. In the case that the mean curvature is non-zero, the smoothness is to be understood in a generalised sense, i.e. also allowing the tangential touching of two smooth CMC hypersurfaces (e.g. two spheres touching).

##### Min-max minimal hypersurfaces with free boundary

I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the min-max method. Our result allows the min-max free boundary minimal hypersurface to be improper; nonetheless the hypersurface is still regular.

##### Multiplicity of constant Q-curvature metrics

I will describe some multiplicity result about constant Q-curvature metrics in the case of homogeneous vibrations and in the case of non compact manifolds related to a singular version of the Q-curvature problem.

##### Quasiregular mappings: smoothness, branching and ellipticity

Quasiregular maps are higher-dimensional analogs of analytic functions, and non-injective generalizations of quasiconformal maps. The first part of the talk will focus on the role of distortion and smoothness in the branching behavior of Euclidean quasiregular maps. I'll sketch the construction (joint with Kaufman and Wu) of $C^{1,\alpha}$ smooth branched quasiregular maps in dimensions four and higher. The concept of quasiregularity extends naturally to Riemannian manifolds, and a Riemannian $n$-manifold is said to be quasiregularly elliptic if it receives a nonconstant quasiregular mapping from $R^n$. Recent developments in analysis in metric spaces motivate the study of quasiregularity in non-Riemannian settings, for instance, the sub-Riemannian Heisenberg group.

##### Some geometric aspects of the Allen-Cahn equation

In this talk I will discuss both local and global properties of the Allen-Cahn equation in closed manifolds.