Alexandrov's Theorem says that a compact constant mean curvature hypersurface embedded in $\mathbb{R}^{n+1}$ must be a round sphere.

What happens when the mean curvature is small, or bounded? (For instance, we require that $\frac{\text{max}| \nabla \tau |}{\text{min} |\tau|},$ with $\tau$ the mean curvature, be small or bounded).

Does this condition place topological restrictions on compact surfaces?

(For context, I am led to this question in because of the Einstein constraint equations, where many people have studied solutions with "near constant" mean curvature.)

notopological restrictions (beyond the initial thing being a smooth manifold) in the "general relativity" question. We can always change the topology of the solution to accommodate. $\endgroup$ – Willie Wong Oct 8 '14 at 11:51