论文标题
歧管的残留物
Residues of manifolds
论文作者
论文摘要
riesz $ z $ - 歧管$ x $的能量是在产品空间$ x \ times x $上的两个点与功率$ z $之间的距离的集成。被认为是复杂变量$ z $的函数，可以通过分析延续将其推广到meromormorphic函数，我们称之为$ x $的meromormormormormormormormormormormormormorormormorormormorormormorormormorormormorormormororphic函数。它在某些负整数上只有简单的杆子。歧管$ x $的残留物是meromormorormormororphic函数的残基。例如，可以作为残基获得$ \ mathbb {r}^3 $中表面的卷和Willmore Energy。 在本文中，我们首先显示了残留物的Möbius不变性在$ z = -2 \ dim x $的封闭submanifold或欧几里得空间中紧凑的身体。我们引入了紧凑型物体和加权残基的相对残基，并表明标量曲率和平均曲率以及尺寸紧凑型物体的欧拉（Euler）特征小于$ 4 $，可以用残基和局部残基来表达。我们研究了$ x $的本地定义功能的差异顺序，这对于获取（全球）残基是欧几里得空间的封闭子手术时所必需的。我们还显示了包容性排斥原则。 残留物似乎类似于通过渐近扩张获得的数量，例如内在体积（Lipschitz杀死曲线），Laplacian的光谱和Graham-Witten Energy。我们证明残留物与它们独立。最终，我们以$ 4 $尺寸的超曲面为$ \ mathbb {r}^5 $引入了\ m不变的主曲率能量，并以残留物，weyl张量和此莫比乌斯不变的曲线曲线能量来表达Graham-Witten Energy。
The Riesz $z$-energy of a manifold $X$ is the integration of the distance between two points to the power $z$ over the product space $X\times X$. Considered as a function of a complex variable $z$, it can be generalized to a meromorphic function by analytic continuation, which we will call the meromorphic energy function of $X$. It has only simple poles at some negative integers. The residues of a manifold $X$ are the residues of the meromorphic energy function. For example, the volume and the Willmore energy for surfaces in $\mathbb{R}^3$ can be obtained as residues. In this paper we first show the Möbius invariance of the residue at $z=-2\dim X$ of a closed submanifold or a compact body in a Euclidean space. We introduce the relative residues for compact bodies and weighted residues, and show that the scalar curvature and the mean curvature as well as the Euler characteristic of compact bodies of dimension less than $4$ can be expressed in terms of residues and local residues. We study the order of differentiaion of a local defining function of $X$ that is necessary to obtain the (global) residues when $X$ is a closed submanifold of a Euclidean space. We also show the inclusion-exclusion principle. Residues appear to be similar to quantities obtained by asymptotic expansion such as intrinsic volumes (Lipschitz-Killing curvatures), spectra of Laplacian, and the Graham-Witten energy. We show that residues are independent from them. Finally we introduce a \M invariant principal curvature energy for $4$-dimensional hypersurfaces in $\mathbb{R}^5$, and express the Graham-Witten energy in terms of the residues, Weyl tensor, and this Möbius invariant principal curvature energy.
