video. If a Surface $S$ is compact and connected, and its Gaussian curvature > 0, and its Mean curvature is constant, then $S$ is a Sphere.

Proved with Hilbert's theorem.

Proof

We start with the fact that if a surface has non-zero Mean curvature everywhere then it is orientable.

This means that the Principal curvatures are well-defined globally, and because of compactness, there is a global maximum for the larger principal curvature $k_2$, but because mean curvature ($\propto k_2+k_1$) is constant, this implies this same point is a minimum of the smaller principal curvature $k_1$. Therefore, we can apply Hilbert's theorem to deduce that it is an Umbilical point. However, with a simple argument, we see that all points are umbilical, and with all the other assumptions, this means that the surface is a Sphere!