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We establish a lower bound on the ADM mass of the time slices of $(n + 1)$-dimensional asymptotically flat standard static spacetimes under the timelike convergence condition. This extends the Minkowski inequality for static vacuum spacetimes, which relates the total mass of a time slice to the area and total mean curvature of its boundary. As a corollary, we obtain the Riemannian Penrose inequality in all dimensions and with rigidity for static manifolds under the TCC., Comment: 15 pages, no figures, comments welcome!

We prove that a proper weak solution $\{ \Omega_{t} \}_{0 \leq t < \infty}$ to inverse mean curvature flow in $\mathbb{H}^{n}$, $3\leq n \leq 7$, is smooth and star-shaped by the time \begin{equation*} T= (n-1) \log \left( \frac{\text{sinh} \left( r_{+} \right)}{ \text{sinh} \left( r_{-} \right)} \right), \end{equation*} where $r_{+}$ and $r_{-}$ are the geodesic out-radius and in-radius of the initial domain $\Omega_{0}$. The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in $\mathbb{R}^{n}$ due to Chow-Gulliver and uses a result of Li-Wei. In addition to this, our methods establish expanding spheres as the only proper weak IMCF on $\mathbb{H}^{n} \setminus \{ 0 \}$ in all dimensions. As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang and De Lima-Girao to outer-minimizing domains $\Omega_{0} \subset \mathbb{H}^{n}$ in dimensions $3 \leq n \leq 7$. From this, we also extend a Penrose-type inequality to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains of $\mathbb{H}^{n}$ in these dimensions., Comment: The lower bound on the support function from the original version was erroneous. The calculation in Section 3 and the resulting estimate have been corrected. Also added a result on proper weak IMCF in punctured hyperbolic space

We prove that equality in the Minkowski inequality for asymptotically flat static manifolds is achieved only by slices of Schwarzschild space. To show this, we establish uniqueness of *quasi-spherical* static metrics: rotationally symmetric regions of Schwarzschild are the only static vacuum metrics which are quasi-spherical with vanishing shear vector. In addition, we observe that the static Minkowski inequality extends to all dimensions for a connected boundary and to asymptotically flat static manifolds of any positive decay order. Altogether, this yields a robust rigidity criterion for Schwarzschild space. Using this criterion, we recover Israel's static black hole uniqueness theorem under this mild decay assumption. Likewise, the uniqueness theorems for photon surfaces and static metric extensions from the prequel extend to all dimensions under these weaker asymptotics. 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 is eventually smooth in an arbitrary asymptotically flat background., Comment: 57 pages. The main rigidity theorem (theorem 1.3) has been strengthened -- the mean-convex boundary assumption may be removed if the manifold is asymptotically flat of order \tau >0 (see Section 2), which yields the complete rigidity statement for these manifolds. Because of the upgraded rigidity statement, we also included a new proof of static black hole uniqueness in section 8

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds $(M^{n}, g)$ with boundary and with dimension $ n < 8$ that was establishedby McCormick. First, we show that any asymptotically flat static $(M^{n},g)$ which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type 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 recent uniqueness results for static photon surfaces and black holes., Comment: 37 pages, no figures. Correction made in the definition of Schwarzschild stability, asymptotic considerations clarified, and higher-dimensional result included for equipotetnial photon surfaces. Final version to appear in TAMS

We give a simple proof to the computation of ADM mass of the static extensions of small spheres in Wiygul \cite{W1, W2}. It makes use of the mass formula $m = \frac{1}{4\pi} \int_{\partial M} \frac{\partial V}{\partial \nu}$ for an asymptotically flat static manifold with boundary.

For an $H>0$ rotationally symmetric embedded torus $N_{0} \subset \mathbb{R}^{3}$, evolved by Inverse Mean Curvature Flow, we show that the total curvature $|A|$ remains bounded up to the singular time $T_{\max}$. We then show convergence of the $N_{t}$ to a $C^{1}$ rotationally symmetric embedded torus $N_{T_{\max}}$ as $t \rightarrow T_{\max}$ without rescaling. Later, we observe a scale-invariant $L^{2}$ energy estimate on any embedded solution of the flow in $\mathbb{R}^{3}$ that may be useful in ruling out curvature blowup near singularities in general., Comment: 15 pages, 2 figures, some minor typos corrected. Final version to appear in Proceedings of the American Mathematical Society

We prove that the Inverse Mean Curvature Flow of a non-star-shaped, mean-convex embedded sphere in $\mathbb{R}^{n+1}$ with symmetry about an axis and sufficiently long, thick necks exists for all time and homothetically converges to a round sphere as $t \rightarrow \infty$. 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 $\mathbb{R}^{n+1}$. 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 $\mathbb{R}^{3}$., Comment: 40 pages, 7 figures. This version includes more exposition on rotationally symmetric immersions, some additional applications, and a simpler example of a non-star-shaped admissible initial surface

We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in $\mathbb{R}^{n+1}$ 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., Comment: 22 pages, 2 figures. Final version to appear in Mathematical Research Letters. Added clarification in Section 3 on how one relates the global variational solution of IMCF to the variational solution induced by the classical flow. The most recent update was to fix some typos and update contact info in the document, but no further edits were made