In revision at Communications in Mathematical Physics. ArXiv.

We establish an inequality relating the surface gravity and topology of a horizon in a 3-dimensional asymptotically locally hyperbolic static space with the geometry at infinity. Equality is achieved only by the Kottler black holes, and this rigidity leads to several new static black hole uniqueness theorems for a negative cosmological constant. First, the ADS-Schwarzschild black hole with critical surface gravity \kappa=\sqrt{-\Lambda} is unique. Second, the toroidal Kottler black holes are unique in the absence of spherical horizons. Third, the hyperbolic Kottler black holes with mass m>0 are unique *if* the generalized Penrose inequality holds for the corresponding class of static spaces. Building on work of Ge-Wang-Wu-Xia, we then use this fact to obtain uniqueness for static ALH graphs with hyperbolic infinities.

This inequality follows from a generalization of the Minkowski inequality in ADS-Schwarzschild space due to Brendle-Hung-Wang. Using optimal coefficients for the sub-static Heintze-Karcher inequality, we construct a new monotone quantity under inverse mean curvature flow (IMCF) in static spaces with {\Lambda} < 0. Another fundamental tool developed in this paper is a regularity theorem for IMCF in asymptotically locally hyperbolic manifolds. Specifically, we prove that a weak solution of IMCF in an ALH 3-manifold with horizon boundary is eventually smooth. This extends the regularity theorem for a spherical infinity due to Shi and Zhu. 

The Journal of Geometric Analysis Volume 35, Article 403 (2025)

We establish a lower bound on the total mass of the time slices of (n + 1)-dimensional asymptotically flat standard static spacetimes under the timelike convergence condition. The inequality can be viewed equivalently as a Minkowski-type inequality in these spaces, i.e. as a lower bound on the total mean curvature of the boundary, and thus extends inequalities from [3], [22], [19], and [10]. Equality is achieved only by slices of Schwarzschild space and is related to the characterization of quasi-spherical static vacuum metrics from [10]. As a notable special case of the main inequality, we obtain the Riemannian Penrose inequality in all dimensions for static spaces under the TCC. 

In revision at Calculus of Variations and Partial Differential Equations, ArXiv.

We prove that a proper weak solution {Ωt}0≤t<∞ to inverse mean curvature flow in Hn, 3 ≤ n ≤ 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 Ω0. The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in R n due to Chow-Gulliver from [8] and uses a result of Li-Wei [24]. In addition to this, our methods establish expanding spheres as the only proper weak IMCF on Hn \ {0} in all dimensions. As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang [4] and De LimaGirao [10] to outer-minimizing domains Ω0 ⊂ Hn in dimensions 3 ≤ n ≤ 7. From this, we also extend the asymptotically hyperbolic Penrose inequality from [9] to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains of Hn in these dimensions. 

Under review, ArXiv.

We prove that equality in the Minkowski inequality for asymptotically flat static manifolds from [McC18] is achieved only by slices of Schwarzschild space. In order to show this, we prove that a static vacuum metric on S n−1 × (r0, ∞) of the form g = u 2 (θ, r)dr2 + r 2 gSn−1 belongs to the Schwarzschild family– that is, we establish uniqueness of quasi-spherical static metrics. Additionally, we strengthen the static Minkowski inequality to hold in all dimensions and under relaxed asymptotic assumptions. Using the complete rigidity statement, we further develop the approach to static uniqueness problems in general relativity from [HW24]. Through this approach, we obtain strengthened versions of the higher-dimensional black hole uniqueness theorem from [AM17] and the photon surface and static metric extension uniqueness theorems from [HW24]. Finally, as a notable by-product of our analysis, we establish regularity of weak inverse mean curvature flow in asymptotically flat manifolds– that is, a weak IMCF eventually becomes smooth in any asymptotically flat background, as was suggested by Huisken-Ilmanen in [HI08]. 

International Mathematics Research Notices, Vol. 2024, No. 2 (2024).

We give a simple proof to the computation of ADM mass of the static extensions of small spheres in Wiygul [13, 14]. It makes use of the mass formula m = 1 4π R ∂M ∂V ∂ν for an asymptotically flat static manifold with boundary. 

Calculus of Variations and Partial Differential Equations 62, No. 125 (2023).

We prove that the Inverse Mean Curvature Flow of a non-star-shaped, meanconvex embedded sphere in R n+1 with symmetry about an axis and sufficiently long, thick necks exists for all time and homothetically converges to a round sphere as t → ∞. Our approach is based on a localized version of the parabolic maximum principle. 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 R n+1. The second is a connection between IMCF and minimal surface theory. Based on previous work by Meeks and Yau in [28] 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 . 

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

For an H > 0 rotationally symmetric embedded torus N0 ⊂ R 3 evolved by Inverse Mean Curvature Flow, we show that the total curvature |A| remains bounded up to the singular time Tmax. This in turn implies convergence of the Nt to a C 1 rotationally symmetric embedded torus NTmax as t → Tmax without rescaling, contrasting sharply with the behavior of other extrinsic flows. Later, we note a scale-invariant L 2 energy estimate on any flow solution in R 3 that may be useful in ruling out curvature blowup near singularities more generally. 

Mathematical Research Letters, Vol. 29, No. 4 (2022), 1065-1086.

We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in R n+1 to become star-shaped. As a consequence, we demonstrate that any connected surface moving by the flow which is not initially a topological sphere develops a singularity or self-intersection within a prescribed time interval depending only on initial data. Finally, we establish the existence of either finite-time singularities or intersections for certain topological spheres under IMCF. 

Proquest, LLC (2021).

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 whichexist 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 InverseMean 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.