Mean curvature of compact surface not homeomorphic to a sphere

Question

I'm using Do Carmo's book to self-study differential geometry, where I encounter Exercise 1 on page 282 (Section 4-5), which talks about the Gaussian curvature of points on the surface.

I'm wondering whether the conclusion can be generalized to mean curvature. For instance, for any compact surface in $\mathbb{R}^{3}$ that's not homeomorphic to a sphere, does there always exist a point with zero mean curvature, i.e $H=0$?

Answer 1

Yes, your hunch is right. Take a very long, thin torus. What do you think about $g\ge 2$?