O'Hara - Residues of manifolds

Abstract

Brylinski’s beta function of a manifold $M$ is a meromorphic function of a complex variable $z$ which is given by the double integral on the product space $M\times M$ of the distance between a pair of points to the power $z$. It has only simple poles, and its residues behave somewhat similarly to magnitudes. I will introduce the relation shown by Gimperlein et.al. and show some properties of residues.

Slides can be found on homepage! https://sites.google.com/site/junohara/home

What are residues

Let $(X,d,\mu )$ be a compact sumbanifold of ${\mathbb{R}}^n$. We define a function $B_X(z)={\int }_{X\times X}d(x,y{)}^{z}d{\mu }_{x}d{\mu }_y\in \mathbb{C}$ we can use geodesigc distance..

The analytic contunuation of the function is meromorphic with only simple poles at some negative integers. It is called the Brylinski Beta function or meromprhis Riesz energy fct of $X$, $B_X$.

Motivation

Problem: Find an ideal representation of a knot such that its energy is minimal. The following energy measures can be used:

• Electrostatic energy of charged knots. (cf. KnotPlot by Rob Scharein). We make a few tweaks to the energy to get. ${\int }_{K\times K}\frac{|x-y{|}^2}{dxdy}$ We then do some regulizations

We can study the functions

• Residues. $R_X(z_0)=Re{s}_{z=z_0}{B}_X$, $z_0$ is a pole of $B_X(z)$.
• Energy: Defined by $E_X(z)=...$

Magnitude

Define the magnitude as usual.

For a compact manifold there are two different ways of obtaining a magnitude. Either

• magntude operator $Z_X(R):u\mapsto Z_{X(Ru(x))}:=\frac{1}{R}\int ...$

How to compute residues

Expectation of the magnitude operator e(t) := t \langle Z_{X}(t)1, 1 \rangle$ = \int_{X\times X} e^{-td(x, y)} … Look at the following example.