The Bernstein Problem in the Heisenberg Group

By Nicola Garofalo, Scott D. Pauls. 2002

76 citations recorded by the source corpus.

We establish the following theorem of Bernstein type for the first Heisenberg group: Let S be a C^2 connected H-minimal surface which is a graph over some plane P, then S is either a non-characteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature condition.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c48e5d3ffbbc19eca8986831aa179e5e26dfa393

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Existence and uniqueness for P-area minimizers in the Heisenberg group (2006, methodology): “Later in [8] the authors obtained a similar Bernsteintype theorem through a different approach.”
Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, background): “We stress that this phenomenon, whose Riemannian counterpart is obviously not true, reflects the fact that H -minimal surfaces are ruled surfaces, see [15,49].”
Minimal surfaces in pseudohermitian geometry (2004, result): “After our paper was completed, we were informed that the above claim had been corrected and some similar results are also obtained in the new preprint ([GP2]).”
Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1 (2006, methodology): “Theorem 5.2 admits the following converse, which is Theorem B in [ GP ].”
Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$ (2008, unknown): “The ’worst’”
Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups , n > 1 (2010, background): “Properties of regular minimal surfaces have been investigated in [24], [31], [8], [7], [23], [19], [2] and [29].”
Constant mean curvature surfaces in sub-Riemannian geometry (2005, unknown): “D G ] 1 7 A ug”
Minimal Surfaces in the Heisenberg Group (2001, methodology): “, how many are there? Having continuous constant Gauss map implies that the surface may not have any characteristic points. Since the submission of this article the author and N. Garofalo have shown ([GP02]) a version of the Bernstein property in the Heisenberg group. 5.2. Uniqueness for the Dirichl”