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 -