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.

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    $\begingroup$ Can you clarify what you mean by a $C^{1}$ estimate for $\Sigma?$ $\endgroup$ – Andy Sanders Feb 13 '19 at 19:31
  • $\begingroup$ 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. $\endgroup$ – user128470 Feb 14 '19 at 9:45

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