Brian Allen

We study the Sobolev stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of a sequence of manifolds $M_i^3$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled for time $t\in [0,T]$. In particular, we consider a sequence of regions of manifolds $U_T^i\subset M_i^3$, foliated by a IMCF, ${\Sigma }_t$, such that if $\partial U_T^i={\Sigma }_0^i\cup {\Sigma }_T^i$ and $m_H({\Sigma }_T^i)\to 0$ then $U_T^i$ converges in $W^{1,2}$ to a flat annulus or in the hyperbolic setting it converges to a annulus portion of hyperbolic space. If instead $m_H({\Sigma }_T^i)-m_H({\Sigma }_0^i)\to 0$ and $m_H({\Sigma }_T^i)\to m>0$ then we show that $U_T^i$ converges in $W^{1,2}$ to a topological annulus portion of the Schwarzschild metric or in the Hyperbolic case to a topological annulus portion of the Anti-de Sitter Schwarzschild metric.

We study the stability of the Positive Mass Theorem (PMT) in the case where a sequence of regions of manifolds with positive scalar curvature $U_T^i\subset M_i^3$ are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF) which may not be uniformly controlled near the boundary. Then if $\partial U_T^i={\Sigma }_0^i\cup {\Sigma }_T^i$, $m_H({\Sigma }_T^i)\to 0$ and extra technical conditions are satisfied we show that $U_T^i$ converges to a flat annulus with respect to Sormani-Wenger Intrinsic Flat (SWIF) convergence.

In this paper we study Inverse Mean Curvature Flow (IMCF) on manifolds that are conformal to a warped product manifold. To this end, we show how the gradient conformal vector field in warped product manifolds is related to the conformal vector field on the conformal metric and use this to gain control of the flow in order to show long time existence. Connections are made to recent results of the author on stability of the positive mass theorem (PMT) and the Riemannian Penrose inequality (RPI) where long time existence of IMCF is an important assumption.

We study the stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of an asymptotically hyperbolic manifold $M^3$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically hyperbolic manifolds $U_T^i\subset M_i^3$, foliated by a smooth solution to IMCF which is uniformly controlled, and if $\partial U_T^i={\Sigma }_0^i\cup {\Sigma }_T^i$ and $m_H({\Sigma }_T^i)\to 0$ then $U_T^i$ converges to a topological annulus portion of hyperbolic space with respect to $L^2$ metric convergence. If instead $m_H({\Sigma }_T^i)-m_H({\Sigma }_0^i)\to 0$ and $m_H({\Sigma }_T^i)\to m>0$ then we show that $U_T^i$ converges to a topological annulus portion of the Anti-deSitter Schwarzschild metric with respect to $L^2$ metric convergence.

We study the stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of an asymptotically flat manifold $M^3$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically flat manifolds $U_T^i\subset M_i^3$, foliated by a smooth solution to IMCF which is uniformly controlled, and if $\partial U_T^i={\Sigma }_0^i\cup {\Sigma }_T^i$ and $m_H({\Sigma }_T^i)\to 0$ then $U_T^i$ converges to a flat annulus with respect to $L^2$ metric convergence. If instead $m_H({\Sigma }_T^i)-m_H({\Sigma }_0^i)\to 0$ and $m_H({\Sigma }_T^i)\to m>0$ then we show that $U_T^i$ converges to a topological annulus portion of the Schwarzschild metric with respect to $L^2$ metric convergence.

In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, we investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in ${\mathbb{H}}^{n+1}$ and show long time existence of the flow as well as asymptotic convergence to horospheres.