%0 Journal Article
%J Journal fur die Reine und Angewandte Mathematik, Issue 671, October 2012, Pages 131-198
%D 2012
%T A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group.
%A Andrea Malchiodi
%A Paul Yang
%A Jih-Hsin Cheng
%A JennFang Hwang
%X In this paper, we study the structure of the singular set for a C 1 smooth surface in the 3-dimensional Heisenberg group ℍ 1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify the Codazzi-like equation by proving a fundamental theorem for local surfaces in ℍ 1
%B Journal fur die Reine und Angewandte Mathematik, Issue 671, October 2012, Pages 131-198
%I SISSA
%G en
%U http://hdl.handle.net/1963/6556
%1 6490
%2 Mathematics
%4 1
%# MAT/05 ANALISI MATEMATICA
%$ Approved for entry into archive by Lucio Lubiana (lubiana@sissa.it) on 2013-03-25T14:33:32Z (GMT) No. of bitstreams: 0
%R 10.1515/CRELLE.2011.159
%0 Journal Article
%J Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177.
%D 2005
%T Minimal surfaces in pseudohermitian geometry
%A Jih-Hsin Cheng
%A JennFang Hwang
%A Andrea Malchiodi
%A Paul Yang
%X We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate some {\em extension} theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space $R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, $C^{2}$ smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold.
%B Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177.
%I Scuola Normale Superiore
%G en
%U http://hdl.handle.net/1963/4579
%1 4347
%2 Mathematics
%3 Functional Analysis and Applications
%4 -1
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-07T09:39:45Z
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%R 10.2422/2036-2145.2005.1.05