Suppose we have a say spherical class $A \neq 0$ in $H_2 (M)$, for $M$ a closed manifold, with $H_2(M)$ non-torsion. Let $S$ be the set of (immersed) class $A$ surfaces in $(M,g)$ with mean curvature bounded from above by a fixed constant $C$. Is the $g$-area function bounded on $S$? 

I preliminary version of the question would be to ask if this holds for loops in $M$, with mean curvature replaced by geodesic curvature, again assuming $H_1(M)$ is non-torsion. But it is surfaces I need.