Gap results for free boundary CMC surfaces in the Euclidean three-ball.

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 annulus 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.