I am looking for an answer to the following question: Let $ (M, g) $ be a complete Riemannian manifold and $ Sigma subset M $ a closed, stable minimal surface. Can you prove? $ C ^ 1 $ Estimates for $ Sigma $ that depends only on $ C ^ 0 $Data of the metric tensor? Does the situation change when you replace stability with area minimization?
I can find several resources that deal with higher-order (curvature) estimates, which are usually more dependent $ g $ (at least $ C ^ 1 $) however.