Vanishing theorems for the cohomology groups of free boundary hypersurfaces

TL;DR

This paper establishes vanishing theorems for the cohomology groups of free boundary hypersurfaces in Euclidean balls, showing that small traceless second fundamental form implies topological simplicity, and also proves a rigidity result for minimal free boundary surfaces.

Contribution

It introduces a universal constant condition for vanishing cohomology of free boundary hypersurfaces and provides a new rigidity result for minimal free boundary surfaces.

Findings

Vanishing of the pth cohomology group under size constraints on the second fundamental form.

Existence of a universal constant C depending on n and p.

Rigidity result for minimal free boundary surfaces in the unit ball.

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 $M^n$ 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 $\mathbb{B}^{2+k}$.