Vanishing theorems for the cohomology groups of free boundary submanifolds

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 pth 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 $$\mathbb {B}^{2+k}$$ .