TY - JOUR

T1 - Instability of graphical strips and a positive answer to the bernstein problem in the heisenberg group ℍ1

AU - Danielli, D.

AU - Garofalo, N.

AU - Nhieu, D. M.

AU - Pauls, S. D.

PY - 2009

Y1 - 2009

N2 - In the first Heisenberg group ℍ1 with its sub-Riemannian struc-ture generated by the horizontal subbundle, we single out a class of C2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deforma-tions for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C2 entire H-minimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in ℍ1 the only stable C2 H- minimal entire graphs, with empty characteristic locus, are the vertical planes.

AB - In the first Heisenberg group ℍ1 with its sub-Riemannian struc-ture generated by the horizontal subbundle, we single out a class of C2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deforma-tions for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C2 entire H-minimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in ℍ1 the only stable C2 H- minimal entire graphs, with empty characteristic locus, are the vertical planes.

UR - http://www.scopus.com/inward/record.url?scp=59849100949&partnerID=8YFLogxK

U2 - 10.4310/jdg/1231856262

DO - 10.4310/jdg/1231856262

M3 - 期刊論文

AN - SCOPUS:59849100949

VL - 81

SP - 251

EP - 295

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 2

ER -