A central question in geometric flows is to understand the formation of singularities. In this talk, I will focus on mean curvature flow, the negative gradient flow of area functional, and explain how the local dynamics influence the shape of the flow and its singular set near a cylindrical singularity, as well as how the topology of the flow changes after passing through such a singularity with generic dynamics. This talk is based on the joint works with Ao Sun and Jinxin Xue.

Mean curvature flow (MCF) is the deformation of surfaces with velocity equal to the mean curvature vector. MCF originated in materials science and is widely used as a tool for geometric and topological problems. Major open questions about MCF include how large of singular sets can form, whether the area of the flow is continuous through singular times, and how the various weak solutions may differ. We address these questions under an assumption on the size of the set of singularities with “slow” mean curvature growth. With this assumption, an n-dimensional flow has H^n-measure zero singular sets at every time, has mass that is continuous through singular times, and under an additional mild condition, the level set flow fattens at the discrepancy time of the inner/outer flow. The key technical development is a generalized Brakke equality, which characterizes the deviation from equality in Brakke’s inequality. This is achieved by developing a worldline analysis of Brakke flow, which allows us to relate the regular parts of the flow at different times and estimate the transport of mass into and out of the singular set. 

Abstract: In their seminal work on the minimal surface system, Lawson and Osserman conjectured that Lipschitz graphs that are critical points of the area functional with respect to outer variations are also critical with respect to domain variations. We will discuss the proof of this conjecture for two-dimensional graphs of arbitrary codimension. This is joint work with J. Hirsch and R. Tione.

Abstract: In joint work with Hao Fang, I introduce and prove the existence of metrics on complex surfaces with split tangent bundle. These metrics are analogous to Calabi-Yau metrics, as they flatten certain holomorphically trivial line bundles adapted to the geometric structure, in this case the splitting. First, we will review the Calabi-Yau theorem in the Kahler setting and some issues with generalizing it to non-Kahler manifolds. Then, I will discuss some machinery — introduced by Streets -- that makes it possible to reduce this problem to the study of a family of non-concave full-nonlinear elliptic PDE. Finally, I will show that these PDE are smoothly solvable and draw some parallels to the twisted Monge-Ampere equation.

Abstract: Intermediate k^th Ricci curvatures are curvature conditions interpolating between Sectional curvature (k=1) and Ricci curvature(k=n-1). In this talk I will give a broad overview of what is and isn't known or expected about spaces admitting such metrics, on both sides of the apparent behavioral breakpoint of k=n/2. As an example, I will sketch the proof of an upcoming result that spaces with positive Ric_2 and some fixed degree of symmetry (say an action by T^10) must satisfy the Hopf conjecture, i.e. have positive Euler characteristic, and that the possible cohomology of the fixed points are very restricted. This is joint work with Lee Kennard and Lawrence Mouillé