Properly embedded and immersed minimal surfaces in the Heisenberg group

By Jih-Hsin Cheng, Jenn-Fang Hwang. Bulletin of the Australian Mathematical Society, 2004

59 citations recorded by the source corpus.

We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We givn an explicit expression for these surfaces. Among band types there is a class of properly embedded p-minimal surfaces of so called helicoid type. We classify all the helicoid type p-minimal surfaces. This class of p-minimal surfaces includes all the entire p-minimal graphs (except contact planes) over any plane. Moreover, we give a necessary and sufficient condition for such a p-minimal surface to have no singular points. For general complete immersed p-minimal surfaces, we prove a half space theorem and give a criterion for the properness.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b928bc69a01369ba70a4ed6b28312a3ec037ab2e

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations (2006, methodology): “The calibration method presented here is a refinement of an unpublished result due to L. Ambrosio in [1].”
Existence and uniqueness for P-area minimizers in the Heisenberg group (2006, background): “In [3] and [2], we considered C(2)-smooth solutions u to (1.”
H-minimal graphs of low regularity in H1 (2006, methodology): “A different appr oach to the study of analogues of the Bernstein problem was completed by Cheng, H wang, Malchiodi and Yang in [5] and [4].”
Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, unknown): “The following is only a partial list of references [1–3,6, 14–16,26,28,43–45,48,49,56,74,84,85,90].”
A Notable Family of Entire Intrinsic Minimal Graphs in the Heisenberg Group which are not Perimeter Minimizing (2006, result): “In particular, the papers [GP], [CHMY] and [ CH ] contain a detailed study of the Bernstein problem in the flrst Heisenberg groupH1, and in more general”
Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$ (2008, methodology): “For other aspects of minimal surfaces in the Heisenberg group, including classification and Bernstein-type results, see [8, 27, 2, 22, 21, 37, 36, 35].”
Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1 (2006, methodology): “We mention that, alternatively, one could use the results independently obtained in [CHMY] and [ CH ].”
Constant mean curvature surfaces in sub-Riemannian geometry (2005, unknown): “D G ] 1 7 A ug”
Regularity of C1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group (2007, background): “Starting from the work [6] (see also [5]), we studied the subject from the viewpoint of partial differential equations and that of differential geometry.”
Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group (2016, background): “This vector is called by some authors characteristic direction and plays an important role in the study of minimal surfaces in H and in the study of the properties of the characteristic set char( ), see for instance [12,13] and [11].”
Minimal surfaces in pseudohermitian geometry (2004, background): “In [ CH ], two of us classify all the entire p-minimal graphs over any plane among other things.”