Area-minimizing ruled graphs and the Bernstein problem in the Heisenberg group
Area-minimizing ruled graphs and the Bernstein problem in the Heisenberg group
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DOI：
10.1007/s00526-022-02264-x
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发表时间：
2022-08-01
影响因子：
2.1
通讯作者：
Young, Robert
中科院分区：
文献类型：
--
作者：
Young, Robert
本文给出了Heisenberg群H中的一个图带在板{-1 < x < 1}中是面积极小的一个充要条件。我们表明，我们的条件是必要的，通过引入一个家庭的变形的图形带的基础上变化的垂直曲线。我们证明了这是充分的，通过显示满足条件的条带有单调epigraphs。我们使用这个条件表明，任何面积最小化规则的整个固有图在海森堡群是一个垂直的平面，并找到一个边界曲线，承认不可数许多填充面积最小化的表面。
In this paper, we give a necessary and sufficient condition for a graphical strip in the Heisenberg group 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.