In this note, we prove that if a free boundary constant mean curvature surface $Sigma$ in an Euclidean 3-ball satisfies a pinching condition on the length of traceless second fundamental tensor, then either $Sigma$ is a totally umbilical disk or an a
nnulus of revolution. The pinching is sharp since there are portions of some Delaunay surfaces inside the unit Euclidean 3-ball which are free boundary and satisfy the pinching condition.
In this paper, we prove that there exists a universal constant $C$, depending only on positive integers $ngeq 3$ and $pleq n-1$, 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 $mathbb{B}^{2+k}$.
In this paper, we consider compact free boundary constant mean curvature surfaces immersed in a mean convex body of the Euclidean space or in the unit sphere. We prove that the Morse index is bounded from below by a linear function of the genus and number of boundary components.
Let $M$ be a compact constant mean curvature surface either in $mathbb{S}^3$ or $mathbb{R}^3$. In this paper we prove that the stability index of $M$ is bounded below by a linear function of the genus. As a by product we obtain a comparison theorem b
etween the spectrum of the Jacobi operator of $M$ and those of Hodge Laplacian of $1$-forms on $M$.
