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Riesz energies and the magnitude of manifolds

Heiko Gimperlein, Magnus Goffeng

TL;DR

The paper links Leinster's magnitude of a compact geometry to Brylinski's beta function via a one-parameter family $\mathcal{M}_X(R,\nu)$ built from the magnitude operator. For compact homogeneous spaces, residues of the beta function coincide with information in the magnitude’s large-$R$ expansion, and, using pseudodifferential methods, this relation extends to general closed manifolds through an interpolating polynomial framework. The theory is demonstrated through explicit analyses of $G/H$ spaces and through concrete examples, including the $p$-adic integers and the sphere, where the precise pole structure and asymptotics are recovered. The results illuminate how magnitude encodes geometric data (volume, curvature) and reveal new properties of magnitude in non-manifold settings. Practically, the work provides a toolkit to translate between Mellin-transform residues and magnitude asymptotics, offering deeper insight into the geometry captured by magnitude.

Abstract

We study the geometric significance of Leinster's magnitude invariant. For closed manifolds we find a precise relation with Brylinski's beta function and therefore with classical invariants of knots and submanifolds. In the special case of compact homogeneous spaces we obtain an elementary proof that the residues of the beta function contain the same geometric information as the asymptotic expansion of the magnitude function. For general closed manifolds we use the recent pseudodifferential analysis of the magnitude operator to relate these via an interpolating polynomial family. Beyond manifolds, the relation with the Brylinski beta function allows to deduce unexpected properties of the magnitude function for the $p$-adic integers.

Riesz energies and the magnitude of manifolds

TL;DR

The paper links Leinster's magnitude of a compact geometry to Brylinski's beta function via a one-parameter family built from the magnitude operator. For compact homogeneous spaces, residues of the beta function coincide with information in the magnitude’s large- expansion, and, using pseudodifferential methods, this relation extends to general closed manifolds through an interpolating polynomial framework. The theory is demonstrated through explicit analyses of spaces and through concrete examples, including the -adic integers and the sphere, where the precise pole structure and asymptotics are recovered. The results illuminate how magnitude encodes geometric data (volume, curvature) and reveal new properties of magnitude in non-manifold settings. Practically, the work provides a toolkit to translate between Mellin-transform residues and magnitude asymptotics, offering deeper insight into the geometry captured by magnitude.

Abstract

We study the geometric significance of Leinster's magnitude invariant. For closed manifolds we find a precise relation with Brylinski's beta function and therefore with classical invariants of knots and submanifolds. In the special case of compact homogeneous spaces we obtain an elementary proof that the residues of the beta function contain the same geometric information as the asymptotic expansion of the magnitude function. For general closed manifolds we use the recent pseudodifferential analysis of the magnitude operator to relate these via an interpolating polynomial family. Beyond manifolds, the relation with the Brylinski beta function allows to deduce unexpected properties of the magnitude function for the -adic integers.
Paper Structure (9 sections, 7 theorems, 42 equations)

This paper contains 9 sections, 7 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Residues of manifolds
  3. Magnitude and complex powers for compact geometries
  4. Homogeneous spaces
  5. Expansion coefficients on smooth manifolds
  6. Examples
  7. $p$-adic integers
  8. Spheres
  9. Asymptotics of Mellin transforms

Key Result

Theorem 1

Let $(X,\mathrm{d},\mu)$ be a compact metric measure space of the following form: Then the function is holomorphic for $\mathrm{Re}(R)\gg 0$ and $\nu\in \mathbb{C}$.

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Proposition 2
  • Theorem 3
  • Remark 4
  • Proposition 5
  • proof
  • Remark 6
  • Theorem 7
  • Theorem 8
  • ...and 2 more