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An Inverse Problem for Renormalized Area: Determining the Bulk Metric with Minimal Surfaces

Jared Marx-Kuo

Abstract

We present an inverse problem which uses the renormalized area functional on minimal submanifolds to recover the expansion of asymptotically hyperbolic, conformally compact metrics which are partially even to high order. We use a rigidity argument to determine the conformal infinity of the metric via the renormalized area. We then consider renormalized volume of perturbations of the hemisphere to determine the higher order terms in the asymptotic expansion of the metric. We prove rigidity when these metrics are log-analytic, and further note that renormalized area determines the obstruction tensor for PE metrics.

An Inverse Problem for Renormalized Area: Determining the Bulk Metric with Minimal Surfaces

Abstract

We present an inverse problem which uses the renormalized area functional on minimal submanifolds to recover the expansion of asymptotically hyperbolic, conformally compact metrics which are partially even to high order. We use a rigidity argument to determine the conformal infinity of the metric via the renormalized area. We then consider renormalized volume of perturbations of the hemisphere to determine the higher order terms in the asymptotic expansion of the metric. We prove rigidity when these metrics are log-analytic, and further note that renormalized area determines the obstruction tensor for PE metrics.
Paper Structure (43 sections, 24 theorems, 248 equations, 4 figures)

This paper contains 43 sections, 24 theorems, 248 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of Results
  4. Acknowledgements
  5. Background
  6. Conformally Compact, Asymptotically Hyperbolic manifolds
  7. Partially Even Metrics
  8. Log-analytic expansions
  9. Renormalized Volume through Riesz Regularization
  10. Minimal Surfaces in AH spaces
  11. Background on Inverse Problems
  12. Preliminaries and Notation
  13. Dilation Maps
  14. Parameterization of hemispheres in hyperbolic space
  15. Mechanics of Finite Part Evaluation
  16. ...and 28 more sections

Key Result

Proposition 1

Let $(M^{n+1}, g)$ and $\gamma \subseteq \partial M$ a $C^{3,\alpha}$ embedded curve. Suppose that $Y^2 \subseteq M^{n+1}$ is a properly embedded minimal surface with asymptotic boundary $\gamma$, then where $\hat{A}$ denotes the trace-free second fundamental form and $W_M$ is the Weyl curvature tensor of $M$.

Figures (4)

  • Figure 1: Visualization of the skew minimal surface over a circle in $\overline{z}$ coordinates
  • Figure 2: Visualization of hemisphere in different coordinate systems.
  • Figure 3: Visualization of $\tilde{Y}_{\gamma, \delta}$
  • Figure 4: Visualization of $Y_t$ over $Y$ and $\gamma_t$ over $\gamma$.

Theorems & Definitions (27)

  • Proposition 1: Alexakis-Mazzeo, Prop 3.1
  • Corollary 1.0.1
  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.2.1
  • Corollary 2.2.2
  • Corollary 2.2.3
  • Corollary 2.2.4
  • Example 1
  • ...and 17 more