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Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface

Pedro Gaspar, Jared Marx-Kuo

TL;DR

The paper addresses the existence of infinitely many hypersurfaces with prescribed mean curvature in manifolds containing a strictly stable minimal surface. It fuses Song’s cylindrical-end construction with Dey’s suspension technique, using a sequence of approximating manifolds and a min-max framework to produce many PMCs of multiplicity one that are almost embedded and disjoint from the contracting neighborhood. By passing to the limit and applying compactness and maximum-principle arguments for PMCs with changing metrics, the authors obtain infinitely many distinct PMCs with area and index bounds tied to the geometry of Σ and to the one-parameter width W_0. The results extend to closed manifolds via metric completion and yield corollaries in the non-Frankel and homology settings, providing partial progress toward the PMC conjecture. Overall, the work advances the multiplicity-one theory for PMCs and demonstrates robust min-max mechanisms in the presence of stable minimal surfaces, with potential implications for the prescribed-mean-curvature landscape on broader geometric backgrounds.

Abstract

We construct infinitely many distinct hypersurfaces with prescribed mean curvature (PMC) for a large class of prescribing functions when $(M^{n+1}, g)$ is a closed smooth manifold containing a minimal surface that is strictly stable (or more generally, admits a contracting neighborhood). In particular, we construct infinitely many distinct PMCs when $H_n(M, \mathbb{Z}_2) \neq 0$, or if $(M, g)$ does not satisfy the Frankel property. Our construction synthesizes ideas from Song's construction of infinitely many minimal surfaces in the non-generic setting, Dey's construction of multiple constant mean curvature surfaces, and Sun--Wang--Zhou's min-max construction of free boundary PMCs.

Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface

TL;DR

The paper addresses the existence of infinitely many hypersurfaces with prescribed mean curvature in manifolds containing a strictly stable minimal surface. It fuses Song’s cylindrical-end construction with Dey’s suspension technique, using a sequence of approximating manifolds and a min-max framework to produce many PMCs of multiplicity one that are almost embedded and disjoint from the contracting neighborhood. By passing to the limit and applying compactness and maximum-principle arguments for PMCs with changing metrics, the authors obtain infinitely many distinct PMCs with area and index bounds tied to the geometry of Σ and to the one-parameter width W_0. The results extend to closed manifolds via metric completion and yield corollaries in the non-Frankel and homology settings, providing partial progress toward the PMC conjecture. Overall, the work advances the multiplicity-one theory for PMCs and demonstrates robust min-max mechanisms in the presence of stable minimal surfaces, with potential implications for the prescribed-mean-curvature landscape on broader geometric backgrounds.

Abstract

We construct infinitely many distinct hypersurfaces with prescribed mean curvature (PMC) for a large class of prescribing functions when is a closed smooth manifold containing a minimal surface that is strictly stable (or more generally, admits a contracting neighborhood). In particular, we construct infinitely many distinct PMCs when , or if does not satisfy the Frankel property. Our construction synthesizes ideas from Song's construction of infinitely many minimal surfaces in the non-generic setting, Dey's construction of multiple constant mean curvature surfaces, and Sun--Wang--Zhou's min-max construction of free boundary PMCs.

Paper Structure

This paper contains 28 sections, 34 theorems, 143 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Main Ideas
  4. Paper Organization
  5. Acknowledgements
  6. Background
  7. Min-max theory
  8. Applications of Min-Max to PMCs
  9. The One Parameter Almgren--Pitts Width of a manifold with boundary
  10. Gluing a cylindrical end
  11. Bounds on $W_0(U_{\epsilon}, g_{\epsilon})$
  12. Approximating sequences $h_{\epsilon} \to h$
  13. Dey's Suspension Construction
  14. Proof of Theorem \ref{['mainTheorem']}
  15. Construction of $Y_{\epsilon,p}$
  16. ...and 13 more sections

Key Result

Theorem 1.1

For $\omega_p(M,g)$ as above on a compact Riemannian manifold, $(M^{n+1}, g)$, there exists $a(n) > 0$ so that

Figures (14)

  • Figure 1: Example of a PMC with large (codimension $0$) touching set
  • Figure 2: Visualization of $N$, the metric completion of $M \backslash \Sigma$
  • Figure 3: Visualization of the constructed PMC, after gluing $\text{Comp}(M\backslash \Sigma)$ back to $M$.
  • Figure 4: Visualization of a dumbell metric which is not Frankel and for which corollary \ref{['mainCorollaryFrankel']} applies.
  • Figure 5: Almost embeddedness can occur on PMCs on large sets where $h \equiv 0$. See zhou2019min
  • ...and 9 more figures

Theorems & Definitions (64)

  • Theorem 1.1: LiokumovichMarquesNevesGuthGromov
  • Theorem 1.2: chodosh2020minimal song2018existence marques2019equidistribution MarquesNevesPositive irie2018density ZhouMultiplicity
  • Conjecture 1.2.1
  • Theorem 1.3: Thm 1.1, DeyCMCs
  • Remark 1.1
  • Theorem 1.4
  • Corollary 1.4.1
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.6.1
  • ...and 54 more