Minimal Surfaces in the Heisenberg Group

By Scott D. Pauls. 2001

133 citations recorded by the source corpus.

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot–Carathéodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot–Carathéodory metric. We generate a large library of examples of minimal surfaces and use these to show that the solution to the Dirichlet problem need not be unique. Moreover, we show that the minimal surfaces we construct are in fact X-minimal surfaces in the sense of Garofalo and Nhieu.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/de2708f02d6f068eceeec3e95b7324021b53ee3a

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

H-minimal graphs of low regularity in H1 (2006, methodology): “The first variation formula has been explored in a variety of settings by a number of authors including the aforementioned paper of Cheng, Hwang, Malchiodi and Yang ([5]), Danielli, Garofalo and Nhieu ([7]), the author ([16]), Bonk and Capogna ([1]), Ritoré and Rosales ([17]).”
A Cortical Based Model of Perceptual Completion in the Roto-Translation Space (2006, background): “The existence of minimal surfaces in a Carnot group on R3 has been recently proved by [9] and [51].”
A Notable Family of Entire Intrinsic Minimal Graphs in the Heisenberg Group which are not Perimeter Minimizing (2006, unknown): “The development of geometric measure theory in sub-Riemannian spaces has received a strong impulse over the past decade, see [Pa1], [Pa2], [CDG], [KR], [E1], [E2], [E3], [Gro], [GN], [Be], [DS], [DGN1], [AK1], [AK2], [CS1], [A], [FSS1], [Ma1], [FSS2], [Ma2], [CMS], [FSS3], [BRS], [DGN4], [DGN5], [LR”
Regularity of C1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group (2007, background): “The p-minimal (or X-minimal, H-minimal) surfaces have been studied extensively in the framework of geometric measure theory (e.g., [9], [8], [ 17 ]).”
The Bernstein Problem in the Heisenberg Group (2002, methodology): “, an H-minimal surface will be a C2 hypersurface whose horizontal mean curvature H, see Deﬁnition 1.1, vanishes everywhere. This notion of a minimal surface was independently introduced in [DGN2] and [Pau], and the present paper was inspired by these works. Of course, the concept of horizontal mean ”
Minimal surfaces in pseudohermitian geometry (2004, methodology): “In the case of a smooth surface in the Heisenberg group, our definitions coincide with those given in [CDG], [DGN], and [ Pau ].”
Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$ (2008, methodology): “The approach to existence of solutions in these papers is based on a-priori estimates for the minimizers of the approximating Riemannian gε perimeter functionals [33] and on solutions of the ”Riemannian” regularized versions of (1.”
Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1 (2006, methodology): “The role of stability in the sub-Riemannian Bernstein problem has been recently highlighted in [ DGN2 ], where the first”
Existence and uniqueness for P-area minimizers in the Heisenberg group (2006, background): “We notice that, for n = 1, Pauls ([16]) proved the existence of a continuous W 1,p minimizer for X (·) under the assumption that the graph of the prescribed boundary function φ satisfies the bounded slope condition (see [9]).”
Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups , n > 1 (2010, methodology): “Since minimal surfaces arise as critical points of the perimenter functional, the variational formulation naturally provides several notions of non regular solutions (see for instance [24], [31] and [7]).”
Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, methodology): “In this paper, given a C2 hypersurface S in a Carnot group G, we introduce a second fundamental form on S adapted to the horizontal subbundle HG, and a geometric notion of mean curvature of S , and we show that the latter coincides with either one of those proposed in [15,46,79], see Propositions 9.”
Convex isoperimetric sets in the Heisenberg group (2006, background): “6) has been studied by several authors under different regularity assumptions (besides the previous references, see also [4–6, 19, 20]).”