Residues of manifolds

TL;DR

This paper explores the residues of the meromorphic energy function of manifolds, revealing their invariance properties, relationships with geometric quantities, and introducing new curvature energies, thus connecting complex analysis with differential geometry.

Contribution

It introduces the concept of residues of manifolds, proves their M"obius invariance, relates them to classical geometric invariants, and defines a new principal curvature energy for 4D hypersurfaces.

Findings

Residues are M"obius invariant at specific poles.

Residues can express scalar curvature, mean curvature, and Euler characteristic.

Residues are independent of intrinsic volumes, Laplacian spectra, and Graham-Witten energy.

Abstract

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 M\"obius 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 M\"obius invariant principal curvature energy.