Mean convex properly embedded [[phi], [??]]-minimal surfaces in [??] - Ma; Ma; dos Santos, Joao Paulo (Revista matemática iberoamericana, 2022)
We establish curvature estimates and a convexity result for mean convex properly embedded [[phi], [??]]-minimal surfaces in [??], i.e., [phi]-minimal surfaces when [phi] depends only on the third coordinate of [??]. Led by the works on curvature estimates for surfaces in 3-manifolds, due to White for minimal surfaces, to Rosenberg, Souam and Toubiana for stable CMC surfaces, and to Spruck and Xiao for stable translating solitons in [??], we use a compactness argument to provide curvature estimates for a family of mean convex [[phi], [??]]-minimal surfaces in [??]. We apply this result to generalize the convexity property of Spruck and Xiao for translating solitons. More precisely, we characterize the convexity of a properly embedded [[phi], [??]]-minimal surface in [??] with non-positive mean curvature when the growth at infinity of [phi] is at most quadratic. 2020 Mathematics Subject Classification: Primary 35J60; Secondary 53C42. Keywords: [phi]-minimal surface, mean convex, area estimates, curvature estimates, convexity.