Mean convex properly embedded $[\varphi,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^3$

We establish curvature estimates and a convexity result for mean convex properly embedded $\[\varphi,\vec{e}{3}]$-minimal surfaces in $\mathbb{R}^3$, i.e., $\varphi$-minimal when $\varphi$ depends only on the third coordinate of $\mathbb{R}^3$. Led by works 3-manifolds, due to White minimal surfaces, Rosenberg, Souam Toubiana stable CMC Spruck Xiao translating solitons we use compactness argument provide family $\mathbb{R}^{3}$. apply this generalize property solitons. More precisely, characterize $\[\varphi,\vec{e}\_{3}]$-minimal surface $\mathbb{R}^{3}$ with non-positive growth at infinity is most quadratic.