Rigidity of Complete Free Boundary Minimal Hypersurfaces in Convex NNSC Manifolds

TL;DR

The paper addresses the rigidity (nonexistence) of complete two-sided stable free boundary minimal hypersurfaces in convex 4-manifolds, focusing on the unit ball $\mathbb{B}^4$. It develops an inductive warped $\theta$-bubble framework, a generalization of Gromov's $\mu$-bubbles and capillary methods, to propagate positivity from boundary geometry into interior end-structure for manifolds with non-negative curvature conditions. Under the hypotheses $\mathrm{Ric}^X_2 \ge 0$, $\mathrm{II}^{\partial X}_2 \ge 0$, and $H_{\partial X} \ge H_0>0$, any complete two-sided stable free boundary minimal hypersurface must be totally geodesic with $\mathrm{Ric}_X(\nu_M,\nu_M)=0$ along $M$ and $\mathrm{I\!I}_{\partial X}(\nu_M,\nu_M)=0$ along $\partial M$, leading to rigidity. Consequently, there is no such immersion in $\mathbb{B}^4$, illustrating a robust inheritance of boundary-positivity into interior rigidity in higher dimensions via warped $\theta$-bubbles.

Abstract

We prove that in the unit ball of $\mathbb{R}^4$, there is no complete two-sided stable free boundary immersion. The result follows from a rigidity theorem of complete free boundary minimal hypersurfaces in complete 4-manifolds with non-negative intermediate Ricci curvature, convex boundary and weakly bounded geometry. The method uses warped $θ$-bubble, a generalization of capillary surfaces.

Theorems & Definitions (43)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Definition 2.2
• Lemma 2.3: chodosh2024complete, Lemma 4.2
• Definition 2.4
• Definition 2.5
• Lemma 2.6: wu2023free, Lemma 3.1
• Lemma 2.7: wu2023free, Lemma 3.3
• Definition 3.1