TY - JOUR

T1 - Sub-Riemannian calculus on hypersurfaces in Carnot groups

AU - Danielli, D.

AU - Garofalo, N.

AU - Nhieu, D. M.

N1 - Funding Information:
* Corresponding author. E-mail addresses: [email protected] (D. Danielli), [email protected] (N. Garofalo), [email protected] (D.M. Nhieu). 1 Supported in part by NSF grants DMS-0002801 and CAREER DMS-0239771. 2 Supported in part by NSF Grant DMS-0300477.

PY - 2007/10/20

Y1 - 2007/10/20

N2 - 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.

AB - 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.

KW - First and second variation of the horizontal perimeter

KW - H-mean curvature

KW - Horizontal Levi-Civita connection

KW - Horizontal second fundamental form

KW - Intrinsic integration by parts

UR - http://www.scopus.com/inward/record.url?scp=34447503514&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2007.04.004

DO - 10.1016/j.aim.2007.04.004

M3 - 期刊論文

AN - SCOPUS:34447503514

SN - 0001-8708

VL - 215

SP - 292

EP - 378

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 1

ER -