Geometry

I'll describe joint work with Karpukhin, Kusner, and McGrath, in which we produce many new families of closed minimal surfaces in S^3 and free boundary minimal surfaces in B^3 via constrained optimization problems for Laplace and Steklov eigenvalues on surfaces. Along the way, I'll highlight…

Steady Kahler-Ricci solitons are eternal solutions of the Kahler-Ricci flow. I will present new examples of such solitons with strictly positive sectional curvature that live on C^n and provide an answer to an open question of H.-D. Cao in complex dimension n>2. This is joint work with Pak-…

We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in R^3. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining our work with results of Brendle and Choi-Haslhofer-Hershkovits-…

Einstein metrics and Ricci solitons are the fixed points of Ricci flow and model the singularities forming. They are also critical points of natural functionals in physics. Their stability in both contexts is a crucial question, since one should be able to perturb away from unstable models.

We will review some recent progress on the general geometric behavior of homologically area-minimizing subvarieties, namely, objects that minimize area with respect to homologous competitors. They are prevalent in geometry, for instance, as holomorphic subvarieties of a Kahler manifold, or as…

Abstract

We introduce a new way to look at level sets of eigenfunctions by viewing the value of the eigenfunction as an independent time variable, with successive level surfaces evolving over time. The evolution obeys a variational principle analogous to mean curvature flow, and this…

We discuss some regularity results for mean curvature flows from smooth hypersurfaces with conical singularities. We then discuss how to use these results to tackle two conjectures of Ilmanen. 

Recently there have been significant developments in how we can think about singularities in minimal submanifolds. I will discuss this circle of ideas, in particular how the new planar frequency function of B. Krummel & N. Wickramasekera allows for a more efficient and refined study of…

A fundamental result about the dynamics and geometry of hyperbolic manifolds is Besson-Courtois-Gallot's entropy inequality. The volume entropy of a Riemannian metric measures the growth rate of geodesic balls in the universal cover. The result says that given a closed hyperbolic manifold (M,g_0…

In their seminal work on the minimal surface system, Lawsonand Osserman conjectured that Lipschitz graphs that are critical pointsof the area functional with respect to outer variations are alsocritical with respect to domain variations. We will discuss the proof ofthis conjecture for two-…