Vanishing theorems for the cohomology groups of free boundary hypersurfaces

Abstract

In this paper, we prove that there exists a universal constant $C$, depending only on positive integers $n\ge 3$ and $p\le n-1$, such that if $M^n$ is a compact free boundary submanifold of dimension $n$ immersed in the Euclidean unit ball ${\mathbb{B}}^{n+k}$ whose size of the traceless second fundamental form is less than $C$, then the $p$th cohomology group of $M^n$ vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball ${\mathbb{B}}^{2+k}$.

Version with improved constants. The authors thank Ezequiel Barbosa for valuable comments on this paper