Sub-Riemannian calculus on hypersurfaces in Carnot groups

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

Research output: Contribution to journalArticlepeer-review

74 Scopus citations


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
Pages (from-to)292-378
Number of pages87
JournalAdvances in Mathematics
Issue number1
StatePublished - 20 Oct 2007


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


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