The bernstein problem for embedded surfaces in the heisenberg group ℍ1

Donatella Danielli, Nicola Garofalo, Duy Minh Nhieu, Scott D. Pauls

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

Abstract

In the paper [13] we proved that the only stable C2 minimal surfaces in the first Heisenberg group ℍ1 which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this paper we extend the result in [13] to C2 complete embedded minimal surfaces in ℍ1 with empty characteristic locus. We prove that every such a surface without boundary must be a vertical plane. This result represents a sub-Riemannian counterpart of the classical theorems of Fischer-Colbrie and Schoen, [16], and do Carmo and Peng, [15], and answers a question posed by Lei Ni.

Original languageEnglish (US)
Pages (from-to)563-594
Number of pages32
JournalIndiana University Mathematics Journal
Volume59
Issue number2
DOIs
StatePublished - 2010
Externally publishedYes

Keywords

  • Bernstein problem
  • Heisenberg group
  • Intrinsic graph
  • Minimal surfaces

ASJC Scopus subject areas

  • Mathematics(all)

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