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# On the Bernstein Problem in the Three-dimensional Heisenberg Group

Published:2015-10-21
Printed: Mar 2016
• Josef F. Dorfmeister,
Fakultät für Mathematik, TU-München, Boltzmann str. 3, D-85747, Garching, Germany
• Jun-ichi Inoguchi,
Department of Mathematical Sciences, Faculty of Science, Yamagata University, Yamagata, 990-8560, Japan
• Shimpei Kobayashi,
Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan
 Format: LaTeX MathJax PDF

## Abstract

In this note we present a simple alternative proof for the Bernstein problem in the three-dimensional Heisenberg group $\operatorname{Nil}_3$ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch-Rosenberg differential.
 Keywords: Bernstein problem, minimal graphs, Heisenberg group, loop groups, spinors
 MSC Classifications: 53A10 - Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53C42 - Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

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