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Free Boundary Stable Minimal Hypersurfaces in Positively Curved 4-Manifolds

Yujie Wu

TL;DR

This work extends rigidity results for stable minimal hypersurfaces to the free-boundary setting in positively curved 4-manifolds by adapting the $\mu$-bubble method under $R\ge 2$ and $\mathrm{Ric}_2\ge 0$, with weakly convex boundary and weakly bounded geometry. The authors develop end-parabolic theory for mixed Dirichlet-Neumann problems, establish a one-end constraint on nonparabolic ends, and use $\mu$-bubbles to obtain diameter and volume-growth controls that force rigidity. Consequently, any complete stable two-sided free boundary minimal immersion in such ambient manifolds is totally geodesic with $\mathrm{Ric}(\eta,\eta)=0$ along $M$ and $A(\eta,\eta)=0$ along $\partial M$, leading to nonexistence in compact positively curved settings with weakly convex boundary; a counterexample shows the convexity assumption is essential. The paper also provides a versatile toolkit for free boundary minimal geometry in 4-manifolds, highlighting the role of boundary convexity and intermediate-curvature conditions in rigidity phenomena.

Abstract

We show that the combination of nonnegative 2-intermediate Ricci Curvature and strict positivity of scalar curvature forces rigidity of two-sided free boundary stable minimal hypersurface in a 4-manifold with bounded geometry and weakly convex boundary. This extends the method of Chodosh-Li-Stryker to free boundary minimal hypersurfaces in ambient manifolds with boundary.

Free Boundary Stable Minimal Hypersurfaces in Positively Curved 4-Manifolds

TL;DR

This work extends rigidity results for stable minimal hypersurfaces to the free-boundary setting in positively curved 4-manifolds by adapting the -bubble method under and , with weakly convex boundary and weakly bounded geometry. The authors develop end-parabolic theory for mixed Dirichlet-Neumann problems, establish a one-end constraint on nonparabolic ends, and use -bubbles to obtain diameter and volume-growth controls that force rigidity. Consequently, any complete stable two-sided free boundary minimal immersion in such ambient manifolds is totally geodesic with along and along , leading to nonexistence in compact positively curved settings with weakly convex boundary; a counterexample shows the convexity assumption is essential. The paper also provides a versatile toolkit for free boundary minimal geometry in 4-manifolds, highlighting the role of boundary convexity and intermediate-curvature conditions in rigidity phenomena.

Abstract

We show that the combination of nonnegative 2-intermediate Ricci Curvature and strict positivity of scalar curvature forces rigidity of two-sided free boundary stable minimal hypersurface in a 4-manifold with bounded geometry and weakly convex boundary. This extends the method of Chodosh-Li-Stryker to free boundary minimal hypersurfaces in ambient manifolds with boundary.
Paper Structure (8 sections, 26 theorems, 71 equations)

This paper contains 8 sections, 26 theorems, 71 equations.

Table of Contents

  1. Introdution
  2. Set Up
  3. Weakly Bounded Geometry
  4. Parabolicity on Manifolds with Noncompact Boundary
  5. At Most One Nonparabolic End
  6. $\mu$-bubble and Almost Linear Volume Growth
  7. Proof of Main Theorem and Necessity of Convexity Assumption
  8. Appendix

Key Result

Theorem 1.1

Consider $(X^4, \partial X)$ a complete Riemannian manifold with weakly convex boundary, $R \geq 2$, $\text{Ric}_2 \geq 0$, and weakly bounded geometry. Then any complete stable two-sided immersion of free boundary minimal hypersurface $(M,\partial M)\hookrightarrow (X,\partial X)$ is totally geodes

Theorems & Definitions (59)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5: ChodoLiStryk2022-CompleteStableMinimal, Lemma 2.2
  • proof
  • ...and 49 more