%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