On the minimal surfaces in Euclidean space with density

Abstract

In this paper,we study the $\varphi-$Lplace-Beltrami operator of a nonparametric  surface in $\mathbb{R}^{3}$ with density $e^{\varphi}$ and  we give prove that $\Delta_{\varphi}X=2H_{\varphi}.N+\nabla\varphi=2HN+ (\nabla \varphi)^{T}$, where $X$ is a vector position of a nonparametric surface $z=f(x^{1},x^{2})$ in  $\mathbb{R}^{3}$ with density $e^{\varphi},$ and $(\nabla\varphi)^{T}$ the component tangent of $\nabla\varphi,$ and we give some results related by the weighted mean curvature.