Differential Geometry, Geometric Analysis; particularly minimal surfaces and mean curvature flow.
Publications and preprints
- Min-max theory for constant mean curvature hypersurfaces. (with Xin Zhou) Preprint.
We develop a min-max theory for constant mean curvature (CMC) hypersurfaces in a smooth closed ambient manifold of dimension at most 7. In particular we are able to prove the existence of a smooth CMC hypersurface with any prescribed value of the mean curvature. Moreover, for nonzero mean curvature the min-max limit has multiplicity 1.
- Moving-centre monotonicity formulae for minimal submanifolds and related equations. To appear in J. Funct. Anal.. Published version.
We prove a monotonicity formula for energy-like functionals on minimal submanifolds, mean curvature flows and other geometric objects, in which the centre may move as the scale increases. For minimal submanifolds this monotonicity implies an area bound conjectured by Alexander-Hoffman-Osserman and first proven by Brendle-Hung.
- First stability eigenvalue of singulol rear minimal hypersurfaces in spheres. Preprint.
We prove that the least eigenvalue of the Jacobi operator on a singular minimal hypersurface in the round (n+1)-sphere is at most -2n, with equality at the Clifford hypersurfaces. This is an extension of a classical result of J. Simons to the singular setting.
- On the entropy of closed hypersurfaces and singular self-shrinkers. Preprint.
We prove that any closed hypersurface of dimension n has entropy at least that of the round n-sphere, confirming a conjecture of Colding-Ilmanen-Minicozzi-White that was previously known only for n at most 6. A key ingredient is our classification of entropy-stable self-shrinkers which are allowed to have a tame singular set; this extends the classification due to Colding-Minicozzi.
- On the rigidity of mean convex self-shrinkers. (with Qiang Guang) To appear in Int. Math. Res. Not. IMRN. Published version.
We prove rigidity theorems for cylinders amongst self-shrinkers which are mean convex on large balls. In particular we remove the bounded curvature hypothesis from the rigidity theorem of Colding-Ilmanen-Minicozzi when the dimension is at most 6, or if the mean curvature is bounded below.
- Rigidity and Curvature Estimates for Graphical Self-shrinkers. (with Qiang Guang) Preprint.
We prove that self-shrinkers that satisfy a certain stability condition on large balls must in fact be hyperplanes. The stability condition includes the case that each connected component of the shrinker inside the ball is a graph, not necessarily over the same hyperplane.
- Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature, J. Topol. Anal., 9, 505-532 (2017). Published version.
We construct metrics on the hemisphere for which the Ricci curvature is bounded below by a positive constant k, yet the boundary is minimal and has first Laplace eigenvalue strictly less than k. The aim of this project was to investigate potential approaches for proving Yau's conjecture on the first eigenvalue of minimal hypersurfaces in spheres.
A list of my papers may be additionally found on the arXiv.
- An inverse phase stability approach to rational materials synthesis. (joint with W. Sun, W. huang, D. Kramer and G. Ceder) In preparation.