Plateau's Problem for intrinsic graphs in the Heisenberg Group

2025/07/23 by Roberto Monti, Monti, Roberto, Giacomo Vianello +1
Mathematics · Computer Science · #advanced mathematical theories #Graph theory and applications #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2507.17714

Abstract

Using a geometric construction, we solve Plateau's Problem in the Heisenberg group ℍ¹ for intrinsic graphs defined on a convex domain D, under a smallness condition either on the boundary ∂ D or on the Lipschitz boundary datum φ: ∂ D → ℝ. The proof relies on a calibration argument. We then apply these techniques to establish a new regularity result for H-perimeter minimizers.

Citations

• Existence and uniqueness of t-graphs of prescribed mean curvature in Heisenberg groups
• Area-minimizing ruled graphs and the Bernstein problem in the Heisenberg group
• The Bernstein problem for (X,Y)-Lipschitz surfaces in three-dimensional sub-Finsler Heisenberg groups
• Harmonic intrinsic graphs in the Heisenberg group
• Lipschitz minimizers for a class of integral functionals under the bounded slope condition
• Area-stationary and stable surfaces of class C¹ in the sub-Riemannian\n Heisenberg group mathbb H¹
• Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups Hⁿ, n>1
• Regularity of non-characteristic minimal graphs in the Heisenberg group H¹
• Regularity of non-characteristic minimal graphs in the Heisenberg group ℍ¹
• Regularity of C¹ smooth surfaces with prescribed p-mean curvature in the Heisenberg group
• The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations
• A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing

Related

• A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing
• Stable minimal graphs in Heisenberg group ℍⁿ
• Lipschitz approximation of almost \mathbb G-perimeter minimizing boundaries in plentiful groups
• The Bernstein problem for Lipschitz intrinsic graphs in the Heisenberg\n group
• Examples of area-minimizing surfaces in the subriemannian Heisenberg group H¹ with low regularity