Vanishing theorems for the cohomology groups of free boundary submanifolds

In this paper, we prove that there exists a universal constant C , depending only on positive integers n ≥ 3 and p ≤ n - 1 , such that if M n is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball 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 B 2 + k .