Rigidity and Gap Phenomena in the Sphere--Ball Correspondence

Abstract

This survey reviews a collection of parallel phenomena between free boundary submanifolds in the Euclidean unit ball and closed submanifolds in the sphere, with particular emphasis on rigidity mechanisms, pinching thresholds, and canonical models. We do not regard the two theories as a unified system in one-to-one correspondence. Rather, we emphasize that in several typical settings -- including low topology, strong pinching, spectral extremality, and symmetry reduction -- the free boundary condition often forces stronger rigidity in the unit ball than in the closed setting. The exposition is organized around six interconnected themes. We first contrast the failure of the spherical Bernstein problem in high dimensions with the dimension-independent rigidity of free boundary minimal disks in the unit ball. We then discuss the parallel roles played by the Clifford torus and the critical catenoid in uniqueness, Morse index, and eigenvalue characterizations. Next, we review the transition from the Lawson--Simons stable currents method to the Bochner--Hardy techniques developed for free boundary problems, summarize pinching and gap theorems driven by the second fundamental form and its traceless part, and outline the linear comparison framework between Morse index and topology in the minimal, constant mean curvature, and weighted settings. Finally, we survey existence results obtained from group actions, isoparametric foliations, and recent equivariant eigenvalue optimization, thereby illustrating both the striking analogies and the essential boundary-driven differences between the closed spherical theory and the free boundary theory in the ball.