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

• Antonio Luis Martínez-Triviño ^[1] ; João Paulo dos Santos ^[2] ; Antonio Martínez
  1. [1] Universidad de Granada

     Universidad de Granada

     Granada, España

     
  2. [2] Universidade de Brasília

     Universidade de Brasília

     Brasil

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 38, Nº 4, 2022, págs. 1349-1370
• Idioma: inglés
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• Resumen

  • We establish curvature estimates and a convexity result for mean convex properly embedded [\varphi,\vec{e}_{3}][φ, e 3 ]-minimal surfaces in \mathbb{R}^3R 3 , i.e., \varphiφ-minimal surfaces when \varphiφ depends only on the third coordinate of \mathbb{R}^3R 3 . 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 \mathbb{R}^3R 3 , we use a compactness argument to provide curvature estimates for a family of mean convex [\varphi,\vec{e}_{3}][φ, e 3 ]-minimal surfaces in \mathbb{R}^{3}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 [\varphi,\vec{e}_{3}][φ, e 3 ]-minimal surface in \mathbb{R}^{3}R 3 with non-positive mean curvature when the growth at infinity of \varphiφ is at most quadratic.