The magnitude and spectral geometry

TL;DR

This paper explores the geometric meaning of magnitude for smooth manifolds, linking it to spectral geometry, and provides asymptotic expansions revealing geometric invariants like volume and curvature.

Contribution

It establishes the well-definedness and meromorphic continuation of magnitude functions for manifolds, and connects their asymptotics to classical geometric quantities.

Findings

Magnitude function admits a meromorphic continuation for large R.

Asymptotic expansion of magnitude relates to volume, surface area, and curvature.

Proves an asymptotic version of the Leinster-Willerton conjecture.

Abstract

We study the geometric significance of Leinster's notion of magnitude for a smooth manifold with boundary of arbitrary dimension, motivated by open questions for the unit disk in $\mathbb{R}^2$. For a large class of distance functions, including embedded submanifolds of Euclidean space and Riemannian manifolds satisfying a technical condition, we show that the magnitude function is well defined for $R\gg 0$ and admits a meromorphic continuation to sectors in $\mathbb{C}$. In the semiclassical limit $R \to \infty$, the magnitude function admits an asymptotic expansion, which determines the volume, surface area and integrals of generalized curvatures. Lower-order terms are computed by black box computer algebra. We initiate the study of magnitude analogues to classical questions in spectral geometry and prove an asymptotic variant of the Leinster-Willerton conjecture.