Rigidity of Complete Free Boundary Minimal Hypersurfaces in Convex NNSC Manifolds

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 $\theta$-bubble, a generalization of capillary surfaces.