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Volume and Area Renormalizations for Conformally Compact Einstein Metrics

C. Robin Graham

TL;DR

This work develops a systematic framework for volume and area renormalization in conformally compact Einstein manifolds and their minimal submanifolds, motivated by the AdS/CFT correspondence. It constructs parity dependent renormalized invariants V (odd boundary dimension) and L (even), and A (odd codimension) with corresponding anomalies K (even codimension), all derived from sharp asymptotic expansions using special defining functions. By relating expansion coefficients to local curvature data and providing explicit low dimensional formulas, it connects geometric invariants to conformal geometry on the boundary and to physical notions such as the Willmore functional and Gauss-Bonnet type expressions. The results illuminate how boundary conformal structure governs renormalized quantities and align with predictions from conformal field theory contexts.

Abstract

This article describes some geometric invariants and conformal anomalies for conformally compact Einstein manifolds and their minimal submanifolds which have recently been discovered via the Anti-de Sitter/Conformal Field Theory correspondence.

Volume and Area Renormalizations for Conformally Compact Einstein Metrics

TL;DR

This work develops a systematic framework for volume and area renormalization in conformally compact Einstein manifolds and their minimal submanifolds, motivated by the AdS/CFT correspondence. It constructs parity dependent renormalized invariants V (odd boundary dimension) and L (even), and A (odd codimension) with corresponding anomalies K (even codimension), all derived from sharp asymptotic expansions using special defining functions. By relating expansion coefficients to local curvature data and providing explicit low dimensional formulas, it connects geometric invariants to conformal geometry on the boundary and to physical notions such as the Willmore functional and Gauss-Bonnet type expressions. The results illuminate how boundary conformal structure governs renormalized quantities and align with predictions from conformal field theory contexts.

Abstract

This article describes some geometric invariants and conformal anomalies for conformally compact Einstein manifolds and their minimal submanifolds which have recently been discovered via the Anti-de Sitter/Conformal Field Theory correspondence.

Paper Structure

This paper contains 4 sections, 4 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Conformally Compact Einstein Metrics
  3. Volume Renormalization
  4. Area Renormalization

Key Result

Lemma 2.1

A metric on $M$ in the conformal infinity of $g_+$ determines a unique defining function $r$ in a neighborhood of $M$ such that $\overline g|_{TM}$ is the prescribed boundary metric and such that $|dr|^2_{\overline g}=1$.

Theorems & Definitions (7)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1