Let $M$ be a closed manifold with non-torsion $\pi\_2$, and $A$ a non-trivial free homotopy class of a map $f: S^2 \to M$.
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 there a $C=C(A)>0$ sufficiently small, so that the $g$-area function is bounded on $S=S(A,C)$? 


A 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 $\pi\_1(M)$ is non-torsion. But it is surfaces I need.



Mean curvature upper bounds and area, or geodesic curvature upper bounds and length
The answer is No. Indeed it is no for spheres of all dimension assuming one can find an immersed class $A$ $S^n$ with mean curvature less than a given $C(A)$. The counterexamples can be generalized from the following example for $n=1$. 


Suppose we have a closed surface $(M,g)$, and a free homotopy class $A$ geodesic $\gamma$ in $M$. Deform $g$ to $g'$ keeping the geodesic $\gamma$ fixed, so that there is a half spherical bump near $\gamma$. That is so that we have an isometric embedding of $$\phi: H =\{ (x,y,z) \in S^2 \subset R^2 | z \geq 0 \} \to (M,g').$$ So that $\phi (1,0,0)$ is close to $\gamma (0)$ and $\phi\_\* (0,1,0)$ is approximately parallel to $\dot {\gamma} (0)$, with respect to the $g'$ Levi-Civita connection. Given this we may "tie" $\gamma$ to multiple covers of the loop $c$, given by the restriction of $\phi$ to the $z=0$ equator. And this will produce our counterexample. It is somewhat more tricky to do this with $C(A)=0$ but I believe it can be done.


