Vanishing theorems for the cohomology groups of free boundary submanifolds

摘要

In this paper, we prove that there exists a universal constant C, depending only on positive integers n3 and pn-1, such that if Mn is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball Bn+k whose size of the traceless second fundamental form is less than C, then the pth cohomology group of Mn vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball B2+k.