The Brylinski beta function of a coaxial layer

TL;DR

This paper extends the Brylinski beta function to coaxial layers on submanifolds in Euclidean space, proving its meromorphic continuation and relating residues to geometric properties like curvature and torsion.

Contribution

It introduces the Brylinski beta function for coaxial layers and establishes its analytic properties, including explicit residue calculations for space curves.

Findings

Beta function has a meromorphic extension with simple poles.

Residues are expressed in terms of curvature and torsion.

Results apply to coaxial layers on submanifolds, especially space curves.

Abstract

In [Pooja Rani and M. K. Vemuri, The Brylinski beta function of a double layer, Differential Geom. Appl. \textbf{92}(2024)], an analogue of Brylinski's knot beta function was defined for a compactly supported (Schwartz) distribution $T$ on Euclidean space. Here we consider the Brylinski beta function of the distribution defined by a coaxial layer on a submanifold of Euclidean space. We prove that it has an analytic continuation to the whole complex plane as a meromorphic function with only simple poles, and in the case of a coaxial layer on a space curves, we compute some of the residues in terms of the curvature and torsion.