Harmonic intrinsic graphs in the Heisenberg group
Harmonic intrinsic graphs in the Heisenberg group
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DOI：
10.2422/2036-2145.202105_054
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发表时间：
2020-12
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影响因子：
--
通讯作者：
Robert Young
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文献类型：
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作者：
Robert Young
$\mathbb{R}^n$中的极小曲面可以用调和函数的图局部逼近，即，函数的临界点的狄利克雷能量，但没有类似的定理是已知的$H$-极小曲面的三维海森堡组$\mathbb{H}$，这是已知的有奇点。在本文中，我们引入了$\mathbb{H}$中曲面的内蕴Dirichlet能量的定义，并研究了这种能量的临界点，我们称之为切触调和图。几乎平坦的区域$H$-极小曲面往往可以近似这样的图。给出了内在Lipschitz图能量最小化的一个标定条件，构造了具有各种奇点的能量最小化图，证明了内在Lipschitz图和分片光滑内在图能量的一个第一变分公式.
Minimal surfaces in $\mathbb{R}^n$ can be locally approximated by graphs of harmonic functions, i.e., functions that are critical points of the Dirichlet energy, but no analogous theorem is known for $H$-minimal surfaces in the three-dimensional Heisenberg group $\mathbb{H}$, which are known to have singularities. In this paper, we introduce a definition of intrinsic Dirichlet energy for surfaces in $\mathbb{H}$ and study the critical points of this energy, which we call contact harmonic graphs. Nearly flat regions of $H$-minimal surfaces can often be approximated by such graphs. We give a calibration condition for an intrinsic Lipschitz graph to be energy-minimizing, construct energy-minimizing graphs with a variety of singularities, and prove a first variation formula for the energy of intrinsic Lipschitz graphs and piecewise smooth intrinsic graphs.