Mean convex properly embedded [φ, (e)over-right-arrow3]-minimal surfaces in R3

We establish curvature estimates and a convexity result for mean convex properly embedded [phi, (e) over right arrow (3)]-minimal surfaces in R-3, i.e., phi-minimal surfaces when phi depends only on the third coordinate of R3. 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 R-3, we use a compactness argument to provide curvature estimates for a family of mean convex [phi, (e) over right arrow (3)]-minimal surfaces in R-3. 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, (e) over right arrow (3)]-minimal surface in R-3 with non-positive mean curvature when the growth at infinity of phi is at most quadratic.