论文标题
海森伯格组中的谐波固有图
Harmonic intrinsic graphs in the Heisenberg group
论文作者
论文摘要
$ \ mathbb {r}^n $中的最小表面可以通过谐波函数的图在本地近似，即，函数是dirichlet能量的关键点，但没有类似的定理以$ h $ h $ h $ minimal以三维的heisenberg group $ heisenberg group $ mathbb $ \ mathbb $ \ mathbb \ y} $ \ y}而闻名。在本文中，我们介绍了$ \ mathbb {h} $中固有的dirichlet能量的定义，并研究了该能量的关键点，我们称之为触点谐波图。这些图通常可以近似于$ h $ - 最小的表面的几乎平坦区域。我们为固有的lipschitz图提供了校准条件，以构建能量最小化，以各种奇异性构建能量最小图，并证明了固有Lipschitz图的能量和分段光滑的固有图的第一个变化公式。
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.
