The Gauss-Bonnet theorem characterizes the topology of surfaces by means of their Gaussian curvature.

Do there exist results characterizing the topology of surfaces embedded in $\mathbf{R}^3$ via their mean curvature?

For example, I find it hard to imagine how a topological 2-sphere could be embedded in $\mathbf{R}^3$ with, for example, everywhere negative mean curvature. However, I am not familiar with any easy results which immediately rule this out. The Gauss-Bonnet theorem seems insufficient here, because the trace of the shape operator is a priori unrelated to its determinant.

  1. Mean curvature depends on the choice of the "unit normal". If you change the orientation (choose the other unit normal), the computed mean curvature for the sphere is everywhere negative.
  2. In any case. Fix $x_0\in \mathbb{R}^3$. Since the embedding $S$ of the two sphere is compact, there is a point $y$ at which $|y - x_0| = \sup_{x\in S} |x - x_0|$. Thus $S\subset \overline{B_{x_0}(|y - x_0|)}$ and $S$ is tangent to $\partial B_{x_0}(|y-x_0|)$ at $y$. So trivially you have a lower bound on the Hessian and that the mean curvature must be positive at $y$ (for the "correct" choice of orientation).
  3. In fact, this has nothing to do with topology. Let $\Sigma$ be any compact surface with no boundary embedded in $\mathbb{R}^3$, the same argument as above shows that both its mean curvature (with appropriate choice of orientation) and scalar curvature must be positive at some point.

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