Sub-Riemannian calculus on hypersurfaces in Carnot groups

D. Danielli, N. Garofalo, D. M. Nhieu

Research output: Contribution to journalArticlepeer-review

62 Scopus citations

Abstract

We develop a sub-Riemannian calculus for hypersurfaces in graded nilpotent Lie groups. We introduce an appropriate geometric framework, such as horizontal Levi-Civita connection, second fundamental form, and horizontal Laplace-Beltrami operator. We analyze the relevant minimal surfaces and prove some basic integration by parts formulas. Using the latter we establish general first and second variation formulas for the horizontal perimeter in the Heisenberg group. Such formulas play a fundamental role in the sub-Riemannian Bernstein problem.

Original languageEnglish (US)
Pages (from-to)292-378
Number of pages87
JournalAdvances in Mathematics
Volume215
Issue number1
DOIs
StatePublished - Oct 20 2007
Externally publishedYes

Keywords

  • First and second variation of the horizontal perimeter
  • H-mean curvature
  • Horizontal Levi-Civita connection
  • Horizontal second fundamental form
  • Intrinsic integration by parts

ASJC Scopus subject areas

  • Mathematics(all)

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