I'm a physicist studying differential geometry for my GR research, and I come up with the following claim (not sure if it's true or not):

For any compact surface $S$ that's not homeomorphic to a sphere, $S$ must have a point with vanishing mean curvature, i.e $H=0$

Is this claim true or not? I know that by applying Gauss-Bonnet theorem, we can prove an analogous claim for the Gaussian curvature $K$