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.