Yale Geometric Analysis Seminar

Abstract: This talk will discuss regularity results for harmonic maps into a certain kind of metric space, called Euclidean buildings.  This will include a discussion of what these spaces are, the behavior of harmonic maps into these spaces, and the connection to superrigidity results for lattices.  We will discuss previously known regularity results before presenting a recent theorem of Breiner, Mese, and myself which generalizes these considerably.  

Quantitative unique continuation states that eigenfunctions of the Laplacian on smooth closed manifolds cannot vanish faster than exponentially in its eigenvalue in any open set. It is interpreted physically that the probability of a quantum particle to appear in the classically forbidden region (total energy < potential) is at least exponentially small, also known as quantum tunnelling. In this informal talk, we will discuss the strategy to prove it on both closed and open manifolds with specified end structures (like cones or cylinders) and discuss open problems. 

 In this talk, we will talk about the proof of that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. As a corollary, it implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. This addresses a question posed by Evans-Spruck and De Giorgi. The talk is based on the joint work with Tang-Kai Lee. 

We formulate a stable Bernstein problem in two types of positively curved manifolds. The problem is completely solved for one type in all dimensions and partially solved for the other in low dimensions, leading to several interesting corollaries. 

The uniqueness of tangent flows is central to understanding singularity formation in geometric flows. A foundational result of Colding and Minicozzi establishes this uniqueness at cylindrical singularities under the Type I assumption in the Ricci flow. In this talk, I will present a strong uniqueness result for cylindrical tangent flows at the first singular time. Our proof hinges on a Łojasiewicz inequality for the pointed $\mathcal{W}$-entropy, which is established under the assumption that the local geometry near the base point is close to a standard cylinder or its quotient. This is joint work with Yu Li.