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تسجيل مستخدم جديدUniqueness of the $[varphi,vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour
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We establish a uniqueness result for the $[varphi,vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour. Well known examples of such cylinders are the grim reaper translating solitons for the mean curvature flow. For such solitons, F. Martin, J. Perez-Garcia, A. Savas-Halilaj and K. Smoczyk proved that, if $Sigma$ is a properly embedded translating soliton with locally bounded genus, and $mathcal{C}^{infty}$-asymptotic to two vertical planes outside a cylinder, then $Sigma$ must coincide with some grim reaper translating soliton. In this paper, applying the moving plane method of Alexandrov together with a strong maximum principle for elliptic operators, we increase the family of $[varphi,vec{e}_{3}]$-minimal graphs where these types of results hold under different assumption of asymptotic behaviour.
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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 surfaces when $varphi$ depends only on the third coordinate of $mathbb{R}^3$. Le
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