@article{7c324025f0e6497b89f53399ba2e1a25,
title = "A Codazzi-like equation and the singular set for C 1 smooth surfaces in the Heisenberg group",
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.",
author = "Cheng, {Jih Hsin} and Hwang, {Jenn Fang} and Andrea Malchiodi and Paul Yang",
note = "Funding Information: Acknowledgment. The authors would like to thank SISSA, Princeton University and Academia Sinica in Taiwan for their kind hospitality. This work has been supported by MiUR, under the project Internazionalizzazione 2005: Equazioni alle derivate parziali nello studio di problemi di analisi geometrica. The third author has been supported by the project FIRB-IDEAS {\textquoteleft}{\textquoteleft}Analysis and Beyond{\textquoteright}{\textquoteright} and by the {\textquoteleft}{\textquoteleft}Giorgio and Elena Petronio Fellowship{\textquoteright}{\textquoteright} while visiting IAS in Princeton in the Fall Semester 2008–2009. Finally we would like to thank the referee for careful reading and valuable comments.",
year = "2012",
month = oct,
doi = "10.1515/CRELLE.2011.159",
language = "English (US)",
pages = "131--198",
journal = "Journal fur die Reine und Angewandte Mathematik",
issn = "0075-4102",
publisher = "Walter de Gruyter GmbH & Co. KG",
number = "671",
}