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Holography and Cheeger constant of asymptotically CMC submanifolds

Samuel Pérez-Ayala, Aaron J. Tyrrell

TL;DR

This work analyzes asymptotically hyperbolic spaces with conformally compact, asymptotically CMC submanifolds to obtain sharp bounds on the Cheeger constant and to connect geometry at infinity with intrinsic/submanifold data. It establishes a universal upper bound on the Cheeger constant in terms of the asymptotic mean curvature and constructs two non-Einstein examples with Cheeger constant $ ext{Ch}=n$ corresponding to positive and negative Yamabe types of the conformal infinity. It also links boundary geometry to global conformal properties by showing that an umbilic boundary characterizes conformally weakly Poincaré--Einstein spaces via vanishing of the third conformal fundamental form, and it introduces an extrinsic conformal invariant obstructing second-order mean-curvature vanishing. In the PE setting, a precise boundary–intrinsic/ extrinsic criterion ties weakly Poincaré--Einstein behavior to vanishing of second and third conformal fundamental forms on the boundary. The paper provides explicit constructions, sharp inequalities, and detailed curvature-expansion machinery (including Fialkow–Gauss relations) that illuminate the interplay between bulk geometry, boundary conformal data, and isoperimetric-type invariants.

Abstract

Let $(M^{n+1},g_+)$ be an asymptotically hyperbolic manifold. We compute the Cheeger constant of conformally compact asymptotically constant mean curvature submanifolds $ ι: Y^{k+1} \to (M^{n+1},g_+)$ with arbitrary codimension. As an application, we provide two classes of examples of $(n+1)$-dimensional asymptotically hyperbolic manifolds with Cheeger constant equal to $n$, whose conformal infinity is of the following types: 1) positive Yamabe invariant, and 2) negative Yamabe invariant. Moreover, in the same spirit as Blitz--Gover--Waldron \cite{BlitzSamuel2021CFFa}, we show that an asymptotically hyperbolic manifold with umbilic boundary is conformally weakly Poincaré--Einstein if and only if the third conformal fundamental form of the boundary vanishes. Next, in the space of asymptotically minimal hypersurfaces $Y$ within a Poincaré--Einstein manifold, we identify an extrinsic conformal invariant of $\partial Y$ which obstructs the vanishing of the mean curvature of $Y$ to second order. This conformal invariant is a linear combination of two Riemannian hypersurface invariants of $\partial Y,$ one which depends on its extrinsic geometry within $\overline{Y}$ and the other on its extrinsic geometry within $\partial M;$ neither of which are conformal invariants individually. Finally, we show that for asymptotically minimal hypersurfaces with mean curvature vanishing to second order inside of a Poincaré--Einstein space, being weakly Poincaré--Einstein is equivalent to the boundary of $Y$ having vanishing second and third conformal fundamental forms when viewed as a hypersurface within the conformal infinity.

Holography and Cheeger constant of asymptotically CMC submanifolds

TL;DR

This work analyzes asymptotically hyperbolic spaces with conformally compact, asymptotically CMC submanifolds to obtain sharp bounds on the Cheeger constant and to connect geometry at infinity with intrinsic/submanifold data. It establishes a universal upper bound on the Cheeger constant in terms of the asymptotic mean curvature and constructs two non-Einstein examples with Cheeger constant corresponding to positive and negative Yamabe types of the conformal infinity. It also links boundary geometry to global conformal properties by showing that an umbilic boundary characterizes conformally weakly Poincaré--Einstein spaces via vanishing of the third conformal fundamental form, and it introduces an extrinsic conformal invariant obstructing second-order mean-curvature vanishing. In the PE setting, a precise boundary–intrinsic/ extrinsic criterion ties weakly Poincaré--Einstein behavior to vanishing of second and third conformal fundamental forms on the boundary. The paper provides explicit constructions, sharp inequalities, and detailed curvature-expansion machinery (including Fialkow–Gauss relations) that illuminate the interplay between bulk geometry, boundary conformal data, and isoperimetric-type invariants.

Abstract

Let be an asymptotically hyperbolic manifold. We compute the Cheeger constant of conformally compact asymptotically constant mean curvature submanifolds with arbitrary codimension. As an application, we provide two classes of examples of -dimensional asymptotically hyperbolic manifolds with Cheeger constant equal to , whose conformal infinity is of the following types: 1) positive Yamabe invariant, and 2) negative Yamabe invariant. Moreover, in the same spirit as Blitz--Gover--Waldron \cite{BlitzSamuel2021CFFa}, we show that an asymptotically hyperbolic manifold with umbilic boundary is conformally weakly Poincaré--Einstein if and only if the third conformal fundamental form of the boundary vanishes. Next, in the space of asymptotically minimal hypersurfaces within a Poincaré--Einstein manifold, we identify an extrinsic conformal invariant of which obstructs the vanishing of the mean curvature of to second order. This conformal invariant is a linear combination of two Riemannian hypersurface invariants of one which depends on its extrinsic geometry within and the other on its extrinsic geometry within neither of which are conformal invariants individually. Finally, we show that for asymptotically minimal hypersurfaces with mean curvature vanishing to second order inside of a Poincaré--Einstein space, being weakly Poincaré--Einstein is equivalent to the boundary of having vanishing second and third conformal fundamental forms when viewed as a hypersurface within the conformal infinity.

Paper Structure

This paper contains 15 sections, 23 theorems, 184 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Discussion of Main Results
  3. Cheeger Constants of Asymptotically CMC Submanifolds
  4. Examples of AH manifolds with and of any sign.
  5. Organization of the Paper.
  6. Preliminaries
  7. Weakly Poincaré--Einstein Manifolds with $\text{Ric}_{g_+}+ng_+\ge 0$
  8. Cheeger Constant on Asymptotically CMC Submanifolds
  9. Examples: Proof of Theorem \ref{['EX-PosNegYamabe']} and Theorem \ref{['WPE-Submanifolds']}
  10. Non-Einstein AH Manifolds with $\text{Ch} = n$ and $\mathcal{Y}>0$.
  11. Non-Einstein AH Manifolds with $\text{Ch} = n$ and $\mathcal{Y}<0$.
  12. WPE Asymptotically Minimal Submanifolds - Proof of Theorem \ref{['WPE-Submanifolds']}
  13. Appendix
  14. Ricci and Scalar curvature of asymptotically minimal hypersurfaces $Y^{n+1}$ in a PE space.
  15. Fialkow--Gauss Equation

Key Result

Theorem 1.1

Assume $n+1> 3$. Let $(M^{n+1},g_+)$ be an AH manifold that satisfies $\text{Ric}_{g_+} + ng_+\ge 0$. Then $g_{(1)} = 0$ and $g_{(2)}=-P_{\hat{g}}$, that is, $g_+$ is WPE if and only if $R_{g_+}+n(n+1) = o(r^2)$, where $r$ is any special defining function and $P_{\hat{g}}$ is the Schouten tensor wit

Figures (3)

  • Figure 1: A conformally compact submanifold $Y^{k+1}$ in an AH space $M^{n+1}$.
  • Figure 2: Holographic coordinates near the boundary of an AH manifold.
  • Figure 3: An asymptotically minimal hypersurface $Y^{n+1}$ sitting inside an AH manifold $M^{n+2}$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Proposition 1.10
  • ...and 33 more