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 $n\geq 3$ and $p\leq n-1$, such that if $M<sup>n$ is a compact free boundary submanifold of dimension $n$ immersed in the Euclidean unit ball $\mathbb{B}<sup>{n+k}$ whose size of the traceless second fundamental form is less than $C$, then the $p$th cohomology group of $M<sup>n$ vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball $\mathbb{B}<sup>{2+k}$.