Research

I specialize in geometric analysis, which lies between differential geometry and partial differential equations and has many applications in physics. I am especially interested in the singular and long-time behavior of geometric flows, geometric inequalities for hypersurfaces, and mathematical problems in general relativity.

Research articles:

    On Weak Inverse Mean Curvature Flow and Minkowski-type Inequalities in Hyperbolic Space

    Preprint. ArXiv

    We prove that a proper weak solution to inverse mean curvature flow in hyperbolic space with 3 \leq n \leq 7 is smooth and star-shaped     by the time T= (n-1)log(sinh (r_{+})/sinh(r_{-})), where r_{+} and r_{-} are the geodesic out-radius and in-radius of the initial domain                \Omega_{0}. 

    As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang and De Lima-Girao to outer-minimizing domains of                 hyperbolic space in these dimensions. From this, we also extend the asymptotically hyperbolic Riemannian Penrose inequality to                 balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains in hyperbolic space.

    Quasi-Spherical Metrics and the Static Minkowski Inequality (joint w/ Y.K. Wang)

    Preprint.  ArXiv

    We prove that equality within the Minkowski inequality for asymptotically flat static spaces is achieved only by slices in Schwarzschild       space for mean-convex, non-negative scalar curvature boundaries. In order to establish this, we prove uniqueness of quasi-spherical           static metrics: any quasi-spherical metric with vanishing shear vector that admits a bounded static vacuum potential is a rotationally          symmetric piece of Schwarzschild space. We also observe that the static Minkowski inequality extends to all dimensions when the             boundary is connected and to weakly asymptotically flat static spaces. As a result, the uniqueness theorems for photon surfaces and        static metric extensions from our prequel extend to this larger class and to all dimensions. 

A Rigidity Theorem for Asymptotically Flat Static Manifolds and its Applications (joint w/ Y.K. Wang)

Transactions of the American Mathematical Society 377 (2024), 3599-3629. ArXiv | Journal

    In this paper, we study the Minkowski inequality for asymptotically flat static manifolds with boundary and with dimension n<8 that was     established by McCormick. First, we show that any asymptotically flat static manifold which achieves the equality and has constant            mean curvature or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we                apply conformal techniques to derive a new Minkowski inequality for the level sets of bounded static potentials. Taken together, these        provide a robust approach to detecting rotational symmetry of asymptotically flat static systems.

    As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild                       coordinate spheres and Euclidean coordinate spheres in dimension n<8 under the natural condition of Schwarzschild stability. This             generalizes an earlier result of Miao. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert                  energy. This is interesting to compare with other uniqueness results for photon surfaces and black holes.

The Mass of the Static Extension of Small Spheres (joint w/ Y.K. Wang)

 International Mathematics Research Notices Volume 2024 Issue 2 (2023), 1606-1623. ArXiv | Journal

We give a simple proof to the computation of the ADM mass of the static extension of small spheres originally by Wiygul. It makes use of the mass formula for an asymptotically flat static manifold with boundary.

Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces

Calculus of Variations and PDE 62 No. 125 (2023). ArXiv | Journal

We prove that the inverse mean curvature flow of a non-star-shaped, mean-convex embedded sphere in Euclidean space with symmetry about an axis and sufficiently long, thick necks exists for all times and homothetically converges to a round sphere as time goes to infinity. We also present two applications of this result. The first is an extension of the Minkowski inequality to the corresponding non-star-shaped, mean-convex domains in Euclidean space. The second is a connection between IMCF and minimal surface theory. Based on previous work by Meeks and Yau and using foliations by IMCF, we establish embeddedness of the solution to Plateau's problem and a finiteness property of stable immersed minimal disks for certain Jordan curves in R^3.

The Limit of the Inverse Mean Curvature Flow on a Torus 

Proceedings of the American Mathematical Society 150 (2022), 3049-3061. ArXiv | Journal

For an H>0 rotationally symmetric embedded torus in R^3 evolved by inverse mean curvature flow, we show that the total curvature |A| remains uniformly bounded up to the singular time. Later, we observe a scale invariant L^2 energy estimate on any closed solution of inverse mean curvature flow in R^3 that may be useful for ruling out curvature blowup near singularities in general.

Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces

Mathematical Research Letters 29 No. 4 (2022), 1065-1086. ArXiv | Journal

We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in Euclidean space to become star-shaped. As a consequence, we demonstrate that any connected surface moving by inverse mean curvature flow without spherical topology develops a singularity or a self-intersection in a prescribed time interval depending only on its initial in-radius and out-radius. Finally, we establish the existence of either finite-time singularities or intersections for certain topological spheres under IMCF. 

My dissertation:

The Inverse Mean Curvature Flow: Singularities, Dynamical Stability, and Applications to Minimal Surfaces. 

Proquest, LLC (2021). Link

This dissertation concerns the Inverse Mean Curvature Flow of closed hypersurfaces in Euclidean Space, and its relationship with minimal surfaces. Inverse Mean Curvature Flow is an extrinsic geometric flow which has become prominent in differential geometry because of its applications to geometric inequalities and general relativity, but deep questions persist about its analytic and geometric structure. The first four chapters of this dissertation focus on singularity formation in the flow, the flow behavior near singularities, and the dynamical stability of round spheres under mean-convex perturbations. On the topic of singularities, I establish the formation of a singularity for all embedded flow solutions which do not have spherical topology within a prescribed time interval. I later show that mean-convex, rotationally symmetric tori undergo a flow singularity wherein the flow surfaces converge to a limit surface without rescaling, contrasting sharply with the singularities of other extrinsic geometric flows. On the topic of long-time behavior, I show that all flow solutions which exist and remain embedded for some minimal time depending only on initial data must exist for all time and asymptotically converge to round spheres at large times. In the fourth chapter, I utilize this characterization to establish dynamical stability of the round sphere under certain mean-convex, axially symmetric perturbations that are not necessarily star-shaped. In the last chapter, I relate questions of singularities and dynamical stability for the Inverse Mean Curvature Flow to the mathematics of soap films. Specifically, I show that certain families of solutions to Plateau’s problem do not self-intersect and remain contained within a given region of Euclidean space. I accomplish this using a barrier method arising from global embedded solutions of Inverse Mean Curvature Flow. Conversely, I also use minimal disks to establish that a singularity likely forms in the flow of a specific mean-convex embedded sphere.