Special Year 2018-19: Variational Methods in Geometry

Abstract: We discuss singularities of Teichmueller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions, and explain in particular how winding singularities of the map component can lead to...

Abstract: Given a class of conformally compact Einstein manifolds with boundary, we are interested to study the compactness of the class under some local and non-local boundary constraints. I will report some joint work with Yuxin Ge and Jie Qing...

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including the mean curvature flow. As an application, we show that an ancient (arbitrarycodimension) mean curvature flow in $S^n$ with low area must...

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...

It has been conjectured that a simply-connected complete Kahler manifold of negatively pinched sectional curvature is biholomorphic to a bounded domain in complex Euclidean space. One evidence is that the manifold is Stein, which is, in particular...

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...

We will describe recent progress on the existence theory and asymptotic analysis for solutions of the complex Ginzburg-Landau equations on closed manifolds, emphasizing connections to the existence of weak minimal submanifolds of codimension two. On...

For an immersed minimal surface in $R^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the...

It is fundamental to understand a manifold with positive scalar curvature and its topology. The minimal surface approach pioneered by R. Schoen and S.T. Yau have advanced our understanding of positively curved manifolds. A very important result is...