Is the following true? I cannot see a counterexample and it seems very intuitively clear, at least in the embedded case.

Claim: Consider the set $S$ of closed immersed Riemann surfaces $\Sigma \subset (X,g)$ (I am particularly interested in spheres), with the magnitude of the mean curvature bounded from above by some $C>0$, where $X$ is also closed. Let $Vol(\Sigma)$ denote the area, and $Diam(\Sigma)$ the diameter. Then for every $\epsilon >0$ there is a $\delta>0$ s.t. if $$Vol (\Sigma) < \delta, \text{ then } Diam (\Sigma)< \epsilon $$ where $\Sigma \in S$.

I really just want to say here that diameter of $\Sigma$ can be assured to be arbitrarily small by requiring that its volume is small.

edit 1: For more clarity let me add that $\epsilon, \delta$ above are assumed to depend on $\Sigma, X, g$.

edit 2: $C$ should be upper bound for magnitude of the mean curvature.

compactsurfaces $\Sigma$ without boundary? Are you making any hypotheses on the ambient Riemannian manifold $(X,g)$? Without some hypotheses such as these, it is hopeless to prove any such estimate, since counterexamples are easily constructed. $\endgroup$ – Robert Bryant Dec 3 '15 at 15:374more comments