Quasi-spherical metrics and the static Minkowski inequality
We prove that equality within the Minkowski inequality for asymptotically flat static manifolds is achieved only by slices of Schwarzschild space.
math.DG↗
arXiv subjects
Brian Harvie
Publications and source records attributed to Brian Harvie.
On the geometry and uniqueness of asymptotically locally hyperbolic static vacuum black holes
We establish several geometric characterizations and rigidity results for 3-dimensional asymptotically locally hyperbolic (ALH) static spaces with horizon boundary. Notably, a 3-dimensional ALH static space with toroidal infinity and strictly non-spherical horizons is isometric to a toroidal Kottler metric. Furthermore, we show that the surface gravity of a static horizon with spherical infinity is bounded below by $\sqrt{3}$, with equality achieved only by the critical AdS-Schwarzschild metric. Consequently, Poincar\'e-Einstein fillings of $S^{2} \times S^{1}(\lambda)$ arising from these spaces have length parameter $\lambda \leq (\sqrt{3})^{-1}$, which supports a recent conjecture of Chang-Yang-Zhang[18]. Finally, static horizons with hyperbolic infinity and non-negative Chru\'sciel-Herzlich mass obey the reverse Riemannian Penrose inequality. In conjunction with work of Ge-Wang-Wu-Xia [29], we use this fact to obtain uniqueness of static ALH graphs with hyperbolic infinities. These results follow from a generalization of the Minkowski inequality in AdS-Schwarzschild space due to Brendle-Hung-Wang [15]. Using optimal coefficients for the sub-static Heintze-Karcher inequality from [24], we construct a new monotone quantity under inverse mean curvature flow (IMCF) in static spaces with negative cosmological constant. Another fundamental tool developed in this paper is a regularity theorem for IMCF in ALH 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-Zhu [52].
math.DG↗
A Penrose-type inequality for static spacetimes
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.
gr-qc↗
On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space
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.
math.DG↗
A rigidity theorem for asymptotically flat static manifolds and its applications
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.
math.DG↗
The mass of the static extension of small spheres
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 $m = \frac{1}{4π} \int_{\partial M} \frac{\partial V}{\partial ν}$ for an asymptotically flat static manifold with boundary.
math.DG↗
The Limit of the Inverse Mean Curvature Flow on a Torus
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.
math.DG↗
Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces
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.
math.DG↗
Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces
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 $\mathbb{R}^{3}$.
math.DG↗