Mean convex properly embedded [φ,e3][\varphi,\vec{e}_{3}]-minimal surfaces in R3\mathbb{R}^3

  • Antonio Martínez

    Universidad de Granada, Spain
  • Antonio Luis Martínez-Triviño

    Universidad de Granada, Spain
  • João Paulo dos Santos

    Universidade de Brasília, Brasília-Df, Brazil
Mean convex properly embedded $[\varphi,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^3$ cover
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We establish curvature estimates and a convexity result for mean convex properly embedded [φ,e3][\varphi,\vec{e}_{3}]-minimal surfaces in R3\mathbb{R}^3, i.e., φ\varphi-minimal surfaces when φ\varphi depends only on the third coordinate of R3\mathbb{R}^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 R3\mathbb{R}^3, we use a compactness argument to provide curvature estimates for a family of mean convex [φ,e3][\varphi,\vec{e}_{3}]-minimal surfaces in R3\mathbb{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 [φ,e3][\varphi,\vec{e}_{3}]-minimal surface in R3\mathbb{R}^{3} with non-positive mean curvature when the growth at infinity of φ\varphi is at most quadratic.

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Antonio Martínez, Antonio Luis Martínez-Triviño, João Paulo dos Santos, Mean convex properly embedded [φ,e3][\varphi,\vec{e}_{3}]-minimal surfaces in R3\mathbb{R}^3. Rev. Mat. Iberoam. 38 (2022), no. 4, pp. 1349–1370

DOI 10.4171/RMI/1352