Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds

TL;DR

The paper proves the existence of compact, almost-embedded free boundary constant mean curvature surfaces with prescribed curvature $c$ in three-dimensional AF manifolds with $R\ge0$ for all $c>0$, and in AH manifolds with $R\ge-6$ for all $c>2$, by marrying min-max theory for the functional $A^c(\Omega)=\mathrm{Area}(\partial\Omega)-c\mathrm{Vol}(\Omega)$ with a robust inverse mean curvature flow (IMCF) regularity framework. A key independent result is that IMCF emerging from a distant point in either AF or AH ends remains smooth up to any prescribed time $T$, provided the end geometry is non-flat, enabling the construction of sweep-outs with favorable isoperimetric properties and ensuring the necessary comparison to Euclidean or hyperbolic models. The authors develop a comprehensive regularity theory for weak IMCF, including a box argument to control star-shapedness and curvature, and extend the hyperbolic case with tailored annulus controls and MCF smoothing. The work then formulates and solves approximate min-max problems in both AF and AH contexts, using recent free boundary min-max theory to produce CMC hypersurfaces and show that the min-max values satisfy $\omega_c(M)<\omega_c(\mathbb{R}^3)$ (AF) or $\omega_c(M)<\omega_c(\mathbb{H}^3)$ (AH), which yields the desired free boundary CMC surfaces. Overall, the paper advances a robust 3D min-max program for CMC surfaces in noncompact geometries and provides a substantial step toward Zhou's conjecture in the noncompact, nonnegative-curvature regime with detailed IMCF regularity in AF and AH ends.

Abstract

We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let $(M^3,g)$ be an asymptotically flat manifold with scalar curvature $R\ge 0$. Then, for each constant $c>0$, there exists a compact, almost-embedded, free boundary constant mean curvature surface $Σ\subset M$ with mean curvature $c$. Likewise, let $(M^3,g)$ be an asymptotically hyperbolic manifold with scalar curvature $R\ge -6$. Then, for each constant $c>2$, there exists a compact, almost-embedded, free boundary constant mean curvature surface $Σ\subset M$ with mean curvature $c$. The proof combines min-max theory with the following fact about inverse mean curvature flow which is of independent interest: for any $T$ the inverse mean curvature flow emerging out of a point $p$ far enough out in an asymptotically flat (or asymptotically hyperbolic) end will remain smooth for all times $t\in (-\infty,T]$.

Figures (2)

• Figure 1: The box argument
• Figure 2: The first pass-through moment $s^*$

Theorems & Definitions (148)

• Conjecture 1.1: X. Zhou
• Remark
• Theorem 1.2: mazurowski2022prescribed
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark
• Theorem 1.7
• Theorem 1.8