Asymptotic and exterior Dirichlet problems for the minimal surface equation in the Heisenberg group with a balanced metric

2019/08/12 by Fidelis Bittencourt, Edson S. Figueiredo, Bittencourt, Fidelis +5
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.1908.04361

openalex publication_date 2019/08/12 · openalex created_date 2019/08/22 · openalex updated_date 2026/07/28

Abstract

It is proved that the Heisenberg group \operatorname*Nil\nolimits₃ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product \mathbbT× Z, where \mathbbT is a totally geodesic surface and ℤ the center of \operatorname*Nil% \nolimits₃. It is then proved the existence of complete properly embedded minimal surfaces in \operatorname*Nil\nolimits₃ by solving the asymptotic Dirichlet problem for the minimal surface equation on \mathbbT. It is also proved the existence of complete properly embedded minimal surfaces foliating an open set of \operatorname*Nil\nolimits₃ having as boundary a given curve Γ in \mathbbT, satisfying the exterior circle condition, by solving the exterior Dirichlet problem for the minimal surface equation in the unbounded connected component of \mathbbT\backslashΓ.

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