@article {2012,
	title = {A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group.},
	journal = {Journal fur die Reine und Angewandte Mathematik, Issue 671, October 2012, Pages 131-198},
	year = {2012},
	publisher = {SISSA},
	abstract = {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},
	doi = {10.1515/CRELLE.2011.159},
	url = {http://hdl.handle.net/1963/6556},
	author = {Andrea Malchiodi and Paul Yang and Jih-Hsin Cheng and JennFang Hwang}
}
@article {2011,
	title = {Axial symmetry of some steady state solutions to nonlinear Schr{\"o}dinger equations},
	journal = {Proc. Amer. Math. Soc. 139 (2011), 1023-1032},
	number = {SISSA;75/2010/M},
	year = {2011},
	publisher = {American Mathematical Society},
	abstract = {In this note, we show the axial symmetry of steady state solutions of nonlinear Schrodinger equations when the exponent of the nonlinearity is between the critical Sobolev exponent of n dimensional space and n - 1 dimensional space.},
	keywords = {Nonlinear Schr{\"o}dinger equation},
	doi = {10.1090/S0002-9939-2010-10638-X},
	url = {http://hdl.handle.net/1963/4100},
	author = {Changfeng Gui and Andrea Malchiodi and Haoyuan Xu and Paul Yang}
}
@article {2005,
	title = {Minimal surfaces in pseudohermitian geometry},
	journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze  (5), 4 (2005) 129-177.},
	number = {arXiv:math/0401136;},
	year = {2005},
	publisher = {Scuola Normale Superiore},
	abstract = {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.},
	doi = {10.2422/2036-2145.2005.1.05},
	url = {http://hdl.handle.net/1963/4579},
	author = {Jih-Hsin Cheng and JennFang Hwang and Andrea Malchiodi and Paul Yang}
}
