Interests

Differential Geometry, Geometric Analysis; particularly minimal surfaces and mean curvature flow.

Publications and preprints

1. Uniqueness of blowups for forced mean curvature flow. (with S. Hirsch) Preprint.

   We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. Our results naturally include the case of mean curvature flows in Riemannian manifolds.

2. Sharp distance comparison for curve shortening flow on the round sphere. (with P. Bryan and M. Langford) Preprint.

   We prove that curve shortening flow on the round sphere displays sharp chord-arc improvement, precisely as in the planar setting. As in the planar case, the sharp estimate implies control on the curvature, resulting in a direct and efficient proof of the long-time behaviour of the flow.

3. A distance comparison principle for curve shortening flow with free boundary. (with M. Langford) Preprint.

   We introduce a reflected chord-arc profile for curves with orthogonal boundary condition and obtain a chord-arc estimate for embedded free boundary curve shortening flows in a convex planar domain. As a consequence, we are able to determine the long-time behaviour of the flow.

4. Moving monotonicity formulae for minimal submanifolds in constant curvature. (with K. Naff) Preprint.

   We discover new monotonicity formulae for minimal submanifolds in space forms, which imply the sharp area bound for minimal submanifolds through a prescribed point in a geodesic ball.

5. The prescribed point area estimate for minimal submanifolds in constant curvature. (with K. Naff) Preprint.

   We prove a sharp area estimate for minimal submanifolds that pass through a prescribed point in a geodesic ball in hyperbolic space. In certain cases, we also prove the corresponding estimate in the sphere. The corresponding Euclidean case was conjectured by Alexander-Hoffman-Osserman and first proven by Brendle-Hung

6. Widths of balls and free boundary minimal submanifolds. Adv. Nonlinear Stud. 23 (2023) Published version.

   We detail certain bounds for the width of constant curvature balls, and investigate related lower bounds for the area of free boundary minimal submanifolds in such balls.

7. Min-max theory for capillary surfaces. (with C. Li and X. Zhou) Preprint.

   We develop a min-max theory for the construction of surfaces with arbitrary constant mean curvature and prescribed boundary contact angle, for generic metrics on an ambient manifold with boundary.

8. Rigidity for spherical product Ricci solitons. (with A. Sun) Preprint.

   We establish quantitative rigidity results for products of two spheres which are shrinking solitons for the Ricci flow. Our results also apply to products of certain Einstein manifolds with a sphere.

9. On certain quantifications of Gromov's non-squeezing theorem. (with K. Sackel, A. Song and U. Varolgunes) To appear in Geometry and Topology.

   We pose the question of how large a set must be removed from a 4-ball of radius larger than 1 in order for it to symplectically embed into a unit cylinder. We find that the Minkowski dmiension of the removed set must be at least 2, and this is optimal when the radius is at most √2. We also discuss the minimal volume that must be removed for the embedding to extend to the whole ball with bounded Lipschitz constant.

10. Łojasiewicz inequalities for mean convex self-shrinkers. Int. Math. Res. Not. IMRN (2021). Published version.

    We apply the perturbative analysis of our previous work to a quantity studied by Colding-Minicozzi on round cylinders. This leads to a somewhat more efficient proof of the Łojasiewicz inequalities for round and Abresch-Langer cylinders.

11. Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers. Preprint.

    We establish explicit Łojasiewicz inequalities for a class of cylindrical self-shrinkers for the mean curvature flow, which includes round cylinders and cylinders over Abresch-Langer curves, and deduce uniqueness of tangent flows and rigidity for such cylinders. The Abresch-Langer case answers a conjecture of Colding-Minicozzi and our proof is new for the round case.

12. Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers. (with A. Sun) Preprint.

    We establish explicit Łojasiewicz inequalities for products of two spheres which are self-shrinkers for the mean curvature flow, and deduce the rigidity of such shrinkers.

13. Mean convex mean curvature flow with free boundary. (with N. Edelen, R. Haslhofer and M. Ivaki) Comm. Pure Appl. Math (2021). Published version.

    We generalise the structure theory of B. White for mean convex mean curvature flows of closed hypersurfaces to flows with free boundary.

14. Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF. (with M. Carmen Domingo-Juan and V. Miquel) J. Differential Equations. 272 (2021), 958-978. Published version.

    We prove a Reilly-type inequality for a drift Laplacian and use it to show that Angenent's shrinking torus is embedded by first eigenfunctions of its drift Laplacian.

15. Min-max theory for networks of constant geodesic curvature. (with Xin Zhou) Adv. Math.. 361 (2020), art. 106941. Published version.

    We extend our min-max theory to construct curves of constant geodesic curvature in any ambient Riemannian surface. The curves may have at most one self-intersection, which is a stationary junction.

16. Existence of hypersurfaces with prescribed mean curvature I - Generic min-max. (with Xin Zhou) Camb. J. Math. 8 (2020), no. 2, 331-362. Published version.

    We consider the problem of finding hypersurfaces whose mean curvature is the restriction of a prescribed function on the ambient manifold. We extend our previous min-max theory to prove the existence of such hypersurfaces for a generic set of smooth prescription functions.

17. Min-max theory for constant mean curvature hypersurfaces. (with Xin Zhou) Invent. Math. 218 (2019), no. 2, 441-490. Published version.

    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.

18. Moving-centre monotonicity formulae for minimal submanifolds and related equations. J. Funct. Anal., 274 (2018), no. 5, 1530-1552. 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.

19. First stability eigenvalue of singular minimal hypersurfaces in spheres. Calc. Var. Partial Differential Equations, 57 (2018), no. 5, art. 130. Published version.

    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.

20. On the entropy of closed hypersurfaces and singular self-shrinkers. J. Differential Geom. 114 (2020), no. 3, 551-593. Published version.

    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.

21. On the rigidity of mean convex self-shrinkers. (with Qiang Guang) Int. Math. Res. Not. IMRN, 2018, no. 20, 6406-6425. 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.

22. Rigidity and Curvature Estimates for Graphical Self-shrinkers. (with Qiang Guang) Calc. Var. Partial Differential Equations, 56 (2017), no. 20, art. 176. Published version.

    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.

23. Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature, J. Topol. Anal., 9 (2017), no.3, 505-532. 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.