2019/02/27 by Yunelsy N Alvarez, Alvarez, Yunelsy N., Ricardo Sá Earp +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1902.10774
openalex publication_date 2019/02/27 · openalex created_date 2022/09/13 · openalex updated_date 2026/07/28
Given a complete n-dimensional Riemannian manifold M, we study the\nexistence of vertical graphs in M\×\ℝ with prescribed mean\ncurvature H=H(x,z). Precisely, we prove that the Dirichlet problem for the\nvertical mean curvature equation in a smooth bounded domain \Ω\⊂ M\nhas solution for arbitrary smooth boundary data if\n(n-1)\H\∂\Ω(y)\≥\nn\sup\z\∈\ℝ\|H(y,z)\| for each\ny\∈\∂\Ω provided the function H also satisfies\n\Riccx\≥ n\sup\z\∈\ℝ ‖\∇x\nH(x,z) ‖- dfracn2n-1\inf\z\∈\ℝ\(H(x,z)\)2\nfor each x\∈\Ω. In the case where M=\ℍn we also establish an\nexistence result if the condition\n\sup\\Ω\×\ℝ\|H(x,z)\|\≤ \(n-1)/(n)\nholds in the place of the condition involving the Ricci curvature. Finally, we\nhave a related result when M is a Hadamard manifold whose sectional curvature\nK satisfies -c2\≤ K\≤ -1 for some c>1. We generalize a classical\nresult of Serrin when the ambient is the Euclidean space.\n