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The structure and rigidiy of FBCMC hypersurfaces

Jia Li

TL;DR

This work addresses rigidity of complete stable two-sided free boundary minimal hypersurfaces in 5-manifolds with boundary under curvature constraints, establishing that strict positivity of the $k$-tri-Ricci curvature together with non-negative $3$-intermediate Ricci curvature forces rigidity in manifolds with weakly convex boundary and bounded geometry. The authors extend Wu's 2023 4D results to dimension five and adapt the free boundary μ-bubble framework alongside 3D diameter estimates to control end behavior and volume growth, culminating in a rigidity conclusion: $M$ is totally geodesic, with $\mathrm{Ric}_N(\eta,\eta)=0$ on $M$ and $A_{\partial N}(\eta,\eta)=0$ on $\partial M$, and the end structure is constrained to at most one non-parabolic end. A corollary of the analysis yields nonexistence of complete stable two-sided free boundary minimal hypersurfaces in compact $(N^5,\partial N)$ with positive sectional curvature and weakly convex boundary. Overall, the paper advances free boundary rigidity theory in higher dimensions by combining μ-bubble variational methods with end-structure analysis under sharp curvature conditions.

Abstract

We prove that the combination of strict positivity of $k$-tri-Ricci curvature with non-negative $3$-intermediate Ricci curvature forces rigidity of two-sided stable free boundary minimal hypersurface in a 5-manifold with bounded geometry and weakly convex boundary. This improves the result of Wu \cite{Wuyujie2023} to 5-dimensions and also extends the method of Hong-Yan \cite{Hong-cmc nonexis} to the free boundary case. We give a characterization of one-endedness for weakly stable free boundary constant mean curvature(FBCMC) hypersurfaces, which is an extension of Cheng-Cheung-Zhou \cite{Cheng-Cheung-Zhou2008}.

The structure and rigidiy of FBCMC hypersurfaces

TL;DR

This work addresses rigidity of complete stable two-sided free boundary minimal hypersurfaces in 5-manifolds with boundary under curvature constraints, establishing that strict positivity of the -tri-Ricci curvature together with non-negative -intermediate Ricci curvature forces rigidity in manifolds with weakly convex boundary and bounded geometry. The authors extend Wu's 2023 4D results to dimension five and adapt the free boundary μ-bubble framework alongside 3D diameter estimates to control end behavior and volume growth, culminating in a rigidity conclusion: is totally geodesic, with on and on , and the end structure is constrained to at most one non-parabolic end. A corollary of the analysis yields nonexistence of complete stable two-sided free boundary minimal hypersurfaces in compact with positive sectional curvature and weakly convex boundary. Overall, the paper advances free boundary rigidity theory in higher dimensions by combining μ-bubble variational methods with end-structure analysis under sharp curvature conditions.

Abstract

We prove that the combination of strict positivity of -tri-Ricci curvature with non-negative -intermediate Ricci curvature forces rigidity of two-sided stable free boundary minimal hypersurface in a 5-manifold with bounded geometry and weakly convex boundary. This improves the result of Wu \cite{Wuyujie2023} to 5-dimensions and also extends the method of Hong-Yan \cite{Hong-cmc nonexis} to the free boundary case. We give a characterization of one-endedness for weakly stable free boundary constant mean curvature(FBCMC) hypersurfaces, which is an extension of Cheng-Cheung-Zhou \cite{Cheng-Cheung-Zhou2008}.

Paper Structure

This paper contains 9 sections, 26 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Organization
  4. PRELIMINARIES
  5. one-endedness of weakly stable FBCMC hypersurfaces
  6. free boundary $\mu$-bubbles and 3-dimensional diameter estimates
  7. Free boundary $\mu$-bubbles
  8. 3-dimensional diameter estimates
  9. rigidity of free boundary minimal hypersurface

Key Result

Theorem 1.1

If $(N^4,g)$ is a complete 4-dimensional Riemannian manifold with weakly bounded geometry, satisfying where $\epsilon_{0}$ is a positive constant, then any complete two-sided stable minimal immersion $M^3\to N^4$ is totally geodesic and has $\mathop{\mathrm{Ric}}\nolimits(\eta,\eta)\equiv 0$ along $M$.

Theorems & Definitions (47)

  • Theorem 1.1: Chodosh-Li-Stryker-4positive curved
  • Theorem 1.2: Hong-cmc nonexis
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Definition 2.1: Shen-Ye
  • Definition 2.2
  • Definition 2.3: Hong-cmc nonexis
  • Definition 2.4
  • ...and 37 more