The Bernstein problem for $(X,Y)$-Lipschitz surfaces in three-dimensional sub-Finsler Heisenberg groups

Abstract

We prove that in the Heisenberg group ${\mathbb{H}}^1$ with a sub-Finsler structure, an $(X,Y)$-Lipschitz surface which is complete, oriented, connected and stable must be a vertical plane. In particular, the result holds for entire intrinsic graphs of Euclidean Lipschitz functions.

In the third version of the manuscript we consider the Bernstein problem for $(X,Y)$-Lipschitz surfaces that are locally intrinsic graphs of Euclidean Lipschitz functions, see Definition 2.2 and Theorem 2.3. We also add Lemma 3.1, Lemma 3.2 and Lemma 5.1 to deal with the low-regularity of $(X,Y)$-Lipschitz surfaces in the first and second variation formulas