I was just wondering, if I have a geodesic ball of radius one in a manifold M whose sectional curvature lies between -epsilon and epsilon for epsilon small, and the injectivity radius of my manifold is two say, and I have a SINGULAR minimal hypersurface in M passing through the centre of my unit ball, can I find a lower bound, away from zero, on the volume of the singular minimal hypersurface contained in the ball, in terms of only the dimension of the ambient manifold M? Thanks!
@Deane, when M is Euclidean space, this is a consequence of monotonicity, which is one of the most fundamental facts about minimal surfaces.
As for the original question, I believe that the answer is YES if the hypersurface S is minimizing in M. Think of S as sitting in the Euclidean ball determined by geodesic coordinates. Of course, S is no longer minimizing in the Euclidean ball, but the curvature bounds should give you control on the ratio of areas computed wrt to the ambient metric vs areas computed wrt the Euclidean metric. So although any "competitor" surface can have Euclidean area less than that of S, it must have area greater than |S|/C for some big constant C. Using a modification of the standard monotonicity argument, this is sufficient to derive a lower bound on the area of S. (For this part, consult arxiv 0705.1128 Section 5 for details.)
As for when S is not necessarily minimizing, I suspect(?) that the answer is still yes. One idea is to look at the proof of Allard's monotonicity formula and account for what happens in the presence of curvature.
I also suspect that all of this is written up in full generality and rigor... somewhere.