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A Gauss-Bonnet formula for the renormalized area of minimal submanifolds of Poincaré-Einstein manifolds

Jeffrey S. Case, C Robin Graham, Tzu-Mo Kuo, Aaron J. Tyrrell, Andrew Waldron

TL;DR

The paper proves a Gauss–Bonnet-type formula for the renormalized area of even-dimensional minimal submanifolds Y in Poincaré–Einstein manifolds X by postulating a submanifold analog of Alexakis' decomposition of conformal invariants. The extrinsic Q-curvature I on Y is conjectured to decompose as I = c Pf + W_Q + div V, which yields renormalized-area expressions A = c_k χ(Y) + ((-1)^{k/2}/(k-1)!) ∫_Y W_Q dA when i:Y→X is polyhomogeneous and the ambient metric is Poincaré–Einstein. The authors verify the conjecture in dimensions k = 2 and k = 4, derive explicit forms for the corresponding W_Q, and connect these with known invariants in the literature; they also provide a renormalized-area formula for minimal submanifolds of Einstein manifolds. The work advances conformal submanifold geometry by producing a robust, invariant integrand for renormalized quantities and ties holographic-type renormalization to intrinsic topological data via the Euler characteristic.

Abstract

Assuming the extrinsic $Q$-curvature admits a decomposition into the Pfaffian, a scalar conformal submanifold invariant, and a tangential divergence, we prove that the renormalized area of an even-dimensional minimal submanifold of a Poincaré-Einstein manifold can be expressed as a linear combination of its Euler characteristic and the integral of a scalar conformal submanifold invariant. We derive such a decomposition of the extrinsic $Q$-curvature in dimensions two and four, thereby recovering and generalizing results of Alexakis-Mazzeo and Tyrrell, respectively. We also conjecture such a decomposition for general natural submanifold scalars whose integral over compact submanifolds is conformally invariant, and verify our conjecture in dimensions two and four. Our results also apply to the area of a compact even-dimensional minimal submanifold of an Einstein manifold.

A Gauss-Bonnet formula for the renormalized area of minimal submanifolds of Poincaré-Einstein manifolds

TL;DR

The paper proves a Gauss–Bonnet-type formula for the renormalized area of even-dimensional minimal submanifolds Y in Poincaré–Einstein manifolds X by postulating a submanifold analog of Alexakis' decomposition of conformal invariants. The extrinsic Q-curvature I on Y is conjectured to decompose as I = c Pf + W_Q + div V, which yields renormalized-area expressions A = c_k χ(Y) + ((-1)^{k/2}/(k-1)!) ∫_Y W_Q dA when i:Y→X is polyhomogeneous and the ambient metric is Poincaré–Einstein. The authors verify the conjecture in dimensions k = 2 and k = 4, derive explicit forms for the corresponding W_Q, and connect these with known invariants in the literature; they also provide a renormalized-area formula for minimal submanifolds of Einstein manifolds. The work advances conformal submanifold geometry by producing a robust, invariant integrand for renormalized quantities and ties holographic-type renormalization to intrinsic topological data via the Euler characteristic.

Abstract

Assuming the extrinsic -curvature admits a decomposition into the Pfaffian, a scalar conformal submanifold invariant, and a tangential divergence, we prove that the renormalized area of an even-dimensional minimal submanifold of a Poincaré-Einstein manifold can be expressed as a linear combination of its Euler characteristic and the integral of a scalar conformal submanifold invariant. We derive such a decomposition of the extrinsic -curvature in dimensions two and four, thereby recovering and generalizing results of Alexakis-Mazzeo and Tyrrell, respectively. We also conjecture such a decomposition for general natural submanifold scalars whose integral over compact submanifolds is conformally invariant, and verify our conjecture in dimensions two and four. Our results also apply to the area of a compact even-dimensional minimal submanifold of an Einstein manifold.
Paper Structure (17 sections, 24 theorems, 252 equations)

This paper contains 17 sections, 24 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Extrinsic $Q$-curvature and scattering compactification
  3. Natural submanifold tensors and extrinsic $Q$-curvature
  4. Asymptotics and scattering compactification
  5. Proofs of Theorems \ref{['main-theorem']} and \ref{['closed case']}
  6. Riemannian and conformal submanifold geometry
  7. Conventions from Riemannian geometry
  8. Riemannian submanifolds
  9. Conformal submanifolds
  10. New conformal submanifold invariants
  11. Identification of invariants
  12. The fourth-order extrinsic $Q$-curvature
  13. Comparison to other conformal submanifold invariants
  14. Proof of \ref{['submanifold-alexakis']} in dimensions two and four
  15. The two-dimensional case
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $k,n \in \mathbb{N}$ with $k$ even and $n > k$. Suppose that eqn:submanifold-decomposition holds for $I=Q$ on $k$-dimensional submanifolds of $n$-manifolds, with $\mathcal{W}_Q$ and $V$ as in the statement of Conjecture submanifold-alexakis. If $(X^{n},g_+)$ is a Poincaré--Einstein manifold, $Y^ where $\mathcal{A}$ denotes the renormalized area of $Y$ and $c_{k}$ is as in Equation eqn:cqygb.

Theorems & Definitions (64)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1: CaseGrahamKuo2023*Theorem 1.1
  • Proposition 2.2: CaseGrahamKuo2023*Theorem 1.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • ...and 54 more