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The Riemannian Penrose Inequality Proved Using Inverse Mean Curvature Flow

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The Riemannian Penrose Inequality Proved Using Inverse Mean Curvature Flow
Wages, Hunter Ray
In this thesis, we discuss the Riemannian Penrose Inequality as well as prove the inequality using monotonicity properties of the solutions of the Inverse Mean Curvature Flow Equation. The proof will follow the process laid out by Huisken and Illmanen in \cite{HandI}. Specifically we will prove that if $M$ is a $n-$dimensional Riemannian Manifold, $m$ is the ADM mass of $M$, and $A(N)$ is the area of the outermost minimal surface of $M$, then\begin{equation} m \geq \sqrt{\frac{A(N)}{16 \pi}}. \end{equation} In order to prove this, we will use properties of the solution to the following Partial Differential Equation: \begin{equation} \pd{x}{t} = \frac{\nu}{H}, \end{equation} where $x \in M$, $\nu$ is the outward pointing unit normal at $x$, and $H$ is the mean curvature. In order to deal with singularities that occur in this PDE, we will perform a level set formulation in order obtain Weak Solutions to Inverse Mean Curvature Flow. This level set formulation follows the process laid out by Evans and Sprunk in \cite{EandS}. When we analyze the weak solutions, we find similar monotonicity properties to the smooth solution case.\\
Differential Geometry
Geometric PDE
Partial Differential Equations
Riemannian Penrose Inequality
Raynor, Sarah G. (committee chair)
Parsley, Jason (committee member)
Moore, Frank (committee member)
2021-06-03T08:36:13Z (accessioned)
2021-06-03T08:36:13Z (available)
2021 (issued)
Mathematics and Statistics (discipline)
http://hdl.handle.net/10339/98821 (uri)
en (iso)
Wake Forest University

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