A Codazzi-like equation and the singular set for C 1 smooth surfaces in the Heisenberg group

Jih Hsin Cheng, Jenn Fang Hwang, Andrea Malchiodi, Paul Yang

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)131-198
Number of pages68
JournalJournal fur die Reine und Angewandte Mathematik
Issue number671
DOIs
StatePublished - Oct 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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