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arXiv · 2105.08890

Area-minimizing ruled graphs and the Bernstein problem in the Heisenberg group

Robert Young
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Abstract

In this paper, we give a necessary and sufficient condition for a graphical strip in the Heisenberg group $\mathbb{H}$ to be area-minimizing in the slab $\{-1<x<1\}$. We show that our condition is necessary by introducing a family of deformations of graphical strips based on varying a vertical curve. We show that it is sufficient by showing that strips satisfying the condition have monotone epigraphs. We use this condition to show that any area-minimizing ruled entire intrinsic graph in the Heisenberg group is a vertical plane and to find a boundary curve that admits uncountably many fillings by area-minimizing surfaces.

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Robert Young. 2021-05-19. Area-minimizing ruled graphs and the Bernstein problem in the Heisenberg group. https://arxiv.org/abs/2105.08890

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