# Reference question: $C^1$ estimate for a stable minimal surface

I am looking for an answer/reference to the following question: Let $$(M,g)$$ be a complete Riemannian manifold and $$\Sigma\subset M$$ a closed, stable minimal surface. Is it possible to prove $$C^1$$ estimates for $$\Sigma$$ that only depend on $$C^0$$-data of the metric tensor? Does the situation change if one replaces stability by area-minimizing?

I'm able to find several resources dealing with higher order (curvature) estimates which usually have a stronger dependence on $$g$$ (at least $$C^1$$) however.

• Can you clarify what you mean by a $C^{1}$ estimate for $\Sigma?$ – Andy Sanders Feb 13 '19 at 19:31
• Let's assume we can cover $M$ by finitely many charts. Then I would like that we can choose local parametrizations such that the $C^1$-norm of these parametrizations w.r.t. to these charts is uniformly bounded. – user128470 Feb 14 '19 at 9:45