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Renormalized Area of Hypersurfaces in Hyperbolic Spaces

Alvaro Pampano

Abstract

We employ Chen's conformal invariant quantity [8, Theorem 1] in combination with the Chern-Gauss-Bonnet formulas to obtain expressions for the renormalized area of asymptotically minimal hypersurfaces in the $(2n+1)$-dimensional hyperbolic space $\mathbb{H}^{2n+1}$, $n=1,2$. Our results extend Alexakis and Mazzeo's formula for the renormalized area for surfaces in $\mathbb{H}^3$ [1, Proposition 3.1] as well as their relation between the renormalized area of minimal surfaces of $\mathbb{H}^3$ and the Willmore energy of their doubles in $\mathbb{R}^3$ [1, Proposition 8.1] to the non-minimal case and to the higher dimensional case $n=2$. Moreover, we also generalize our results by considering hypersurfaces in $(2n+1)$-dimensional Poincaré-Einstein spaces and even-dimensional submanifolds of arbitrary codimension.

Renormalized Area of Hypersurfaces in Hyperbolic Spaces

Abstract

We employ Chen's conformal invariant quantity [8, Theorem 1] in combination with the Chern-Gauss-Bonnet formulas to obtain expressions for the renormalized area of asymptotically minimal hypersurfaces in the -dimensional hyperbolic space , . Our results extend Alexakis and Mazzeo's formula for the renormalized area for surfaces in [1, Proposition 3.1] as well as their relation between the renormalized area of minimal surfaces of and the Willmore energy of their doubles in [1, Proposition 8.1] to the non-minimal case and to the higher dimensional case . Moreover, we also generalize our results by considering hypersurfaces in -dimensional Poincaré-Einstein spaces and even-dimensional submanifolds of arbitrary codimension.
Paper Structure (9 sections, 13 theorems, 123 equations)

This paper contains 9 sections, 13 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Basic Definitions and Notation
  3. Formula for the Area
  4. Renormalized Area
  5. Surfaces in $\mathbb{H}^3$
  6. Hypersurfaces in $\mathbb{H}^5$
  7. Hypersurfaces in Poincaré-Einstein Manifolds
  8. Submanifolds of Arbitrary Codimension
  9. Appendix: Technical Lemmas

Key Result

Proposition 3.2

Let $M$ be a compact hypersurface in $\mathbb{H}^{2n+1}$ with mean curvature $H$, scalar curvature $\lambda$, and extrinsic scalar curvature $R$. The area of $M$ satisfies where $\overline{H}$, $\overline{\lambda}$, and $\overline{R}$ are the mean, the scalar, and the extrinsic scalar curvatures of $M$ regarded as a hypersurface in $\mathbb{R}^{2n+1}$ and $d\overline{A}$ is the volume element on

Theorems & Definitions (18)

  • Remark 3.1
  • Proposition 3.2
  • Definition 4.1
  • Theorem 4.2
  • Remark 4.3
  • Corollary 4.4
  • Theorem 4.5
  • Remark 4.6
  • Corollary 4.7
  • Proposition 5.1
  • ...and 8 more