Geometry

We prove existence for many examples of shrinkers by producing compact, smoothly embedded surfacesthat, under mean curvature flow, develop singularities at which the shrinkers occur as blowups. This is joint work with Paco Martin and Brian White.

We will introduce the half-volume spectrum, a new variational invariant inspired by the classical volume spectrum, but defined by considering only hypersurfaces that divide the ambient manifold into equal volumes. We will present a new min-max theory for constant mean curvature (CMC)…

The $\mathbb{Z}_{2}$-harmonic $1$-forms arises in various compactification problems in gauge theory, deformation problem in special holonomy and calibrated geometry, including those involving flat $PSL(2,\mathbb{C})$ connections, Hitchin equation, Fueter sections, branched deformations of…

In this talk, we will discuss solutions of the inverse mean curvature flow that satisfy the following boundary condition: each hypersurface stays tangential to the boundary of the ambient domain. We will also discuss some recently discovered connections between this class of solutions and scalar…

In recent years, the combined work of Guaraco, Hutchinson, Tonegawa, and Wickramasekera has established a min-max construction of minimal hypersurfaces in closed Riemannian manifolds, based on the analysis of singular limits of sequences of solutions of the Allen—Cahn equation, a semi-linear…

The bridge principle is the idea that you can join compact minimal submanifolds along their boundaries to produce an “approximately minimal” submanifold, called the approximate solution, and apply a small normal perturbation to make the new configuration minimal. It dates back to Lévy (1948…

We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow…

We study the limiting case $\gamma\to(1/2)^-$ in dimension one for the fractional Caffarelli-Kohn-Nirenberg inequality, obtaining Onofri's inequality in the unit disk as a limit. An important aspect is the study of solutions of the weighted Liouville equation for the half-Laplacian in dimension…

Suppose we are given a globally hyperbolic spacetime (M,g) solving the Einstein vacuum equations, and a timelike geodesic in M. I will explain how to construct, on any compact subset of M, a solution g_\epsilon of the Einstein vacuum equations which is approximately equal to g far from the…

I will prove Gromov's conjecture that every 3-manifold of positive scalar curvature contains a short closed geodesic. The proof uses Min-Max theory of minimal surfaces and a combinatorial version of mean curvature flow. Time permitting, I will describe other results about geometry and…