Instability of graphical strips and a positive answer to the bernstein problem in the heisenberg group h

D. Danielli Purdue University N. Garofalo Purdue University D.M. Nhieu San Diego Christian College S.D. Pauls Dartmouth College

Differential Geometry mathscidoc:1609.10094

Journal of Differential Geometry, 81, (2), 251-295, 2009
In the ¯rst Heisenberg group H1 with its sub-Riemannian structure 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 deformations 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 Hminimal 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 Hminimal entire graphs, with empty characteristic locus, are thevertical planes.
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@inproceedings{d.2009instability,
  title={INSTABILITY OF GRAPHICAL STRIPS AND A POSITIVE ANSWER TO THE BERNSTEIN PROBLEM IN THE HEISENBERG GROUP H},
  author={D. Danielli, N. Garofalo, D.M. Nhieu, and S.D. Pauls},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909143813628067751},
  booktitle={Journal of Differential Geometry},
  volume={81},
  number={2},
  pages={251-295},
  year={2009},
}
D. Danielli, N. Garofalo, D.M. Nhieu, and S.D. Pauls. INSTABILITY OF GRAPHICAL STRIPS AND A POSITIVE ANSWER TO THE BERNSTEIN PROBLEM IN THE HEISENBERG GROUP H. 2009. Vol. 81. In Journal of Differential Geometry. pp.251-295. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909143813628067751.
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