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Mountain Pass Critical Points of the Volume Constrained Area Functional

Gregory R. Chambers, Jared Marx-Kuo

TL;DR

The paper develops a mountain-pass min-max framework for the volume-constrained perimeter problem, producing smooth almost-embedded hypersurfaces with nonzero constant mean curvature under generic metrics. Central to the approach are the F^{h,f} and E^f functionals, whose variational structure yields volume-preserving critical points via a constrained min-max theory modeled on Mazurwoski–Zhou and its regularity theory. A key technical advance is the path-connectedness of the smooth Cacciopoli-set space in the old-F topology, established through transversality and a carving lemma, which enables robust construction of constrained sweepouts. The results extend the min-max program for isoperimetric-type problems to a volume-constrained setting, providing new existence results for CMC hypersurfaces and linking variational methods with GMT regularity in a constrained-volume context.

Abstract

We construct mountain pass critical points of the perimeter functional on sets of fixed volume. For a generic metric, this gives rise to a smooth almost embedded hypersurface with non-zero constant mean curvature. Our work utilizes recent techniques of Mazurwoski--Zhou \cite{mazurowski2024infinitely}, and a new result on the connectedness of Cacciopoli sets: any two smooth Cacciopoli sets can be connected by an $\mathbf{F}$-continuous map.

Mountain Pass Critical Points of the Volume Constrained Area Functional

TL;DR

The paper develops a mountain-pass min-max framework for the volume-constrained perimeter problem, producing smooth almost-embedded hypersurfaces with nonzero constant mean curvature under generic metrics. Central to the approach are the F^{h,f} and E^f functionals, whose variational structure yields volume-preserving critical points via a constrained min-max theory modeled on Mazurwoski–Zhou and its regularity theory. A key technical advance is the path-connectedness of the smooth Cacciopoli-set space in the old-F topology, established through transversality and a carving lemma, which enables robust construction of constrained sweepouts. The results extend the min-max program for isoperimetric-type problems to a volume-constrained setting, providing new existence results for CMC hypersurfaces and linking variational methods with GMT regularity in a constrained-volume context.

Abstract

We construct mountain pass critical points of the perimeter functional on sets of fixed volume. For a generic metric, this gives rise to a smooth almost embedded hypersurface with non-zero constant mean curvature. Our work utilizes recent techniques of Mazurwoski--Zhou \cite{mazurowski2024infinitely}, and a new result on the connectedness of Cacciopoli sets: any two smooth Cacciopoli sets can be connected by an -continuous map.

Paper Structure

This paper contains 14 sections, 14 theorems, 66 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Statement of Main Results
  3. Connectedness of $\mathcal{C}^{\infty}(M)$
  4. Perimeter Mountain Pass
  5. Preliminaries
  6. Geometric Measure Theory
  7. Acknowledgements
  8. Connectedness of Cacciopoli Sets
  9. Proof of Thm \ref{['pathConnectedThm']}
  10. Volume Constrained Min-Max for Perimeter
  11. $F^{h,f}$ and $E^f$ functionals
  12. Relative Homotopy Classes
  13. Proof of Theorem \ref{['MPassThm']}
  14. Existence of a Mountainpass

Key Result

Theorem 1.1

Given $(M^{n+1}, g)$ a closed manifold and $3 \leq n+1 \leq 7$, there exists infinitely many distinct, smooth, minimal hypersurfaces.

Figures (7)

  • Figure 1: Continuity in $\mathcal{F}$ does not imply continuity in $\mathbf{F}$.
  • Figure 2: Carving around $A \cap \partial B$ via $\sigma(t)$
  • Figure 3: $S_A$, the resulting set, visualized after smoothing. Note that $\sigma(1) = S_A \sqcup B$.
  • Figure 4: Desingularization of manifold with corner
  • Figure 5: $\sigma_1(t)$ visualized
  • ...and 2 more figures

Theorems & Definitions (25)

  • Theorem 1.1: marques2017existenceirie2018densitychodosh2020minimalsong2023existence
  • Theorem 1.2: Thm 1.1 mazurowski2024infinitely
  • Theorem 1.3: de2018min
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 3.1: Generic Transversality
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3: Carving Lemma
  • ...and 15 more