The magnitude and spectral geometry
Heiko Gimperlein, Magnus Goffeng, Nikoletta Louca
TL;DR
The work establishes a unified semiclassical framework for Leinster's magnitude on smooth manifolds with boundary, using the operator Z_X(R) to recast mag(X,Rd) as a functional of a pseudodifferential parametrix. It proves that, under MR/SMR conditions, M_X(R) has a meromorphic continuation and a large-R asymptotic M_X(R) ~ (1/(n! ω_n)) ∑ c_j(X) R^{n−j}, where c_0 equals the volume and c_1, c_2 encode boundary area and curvature integrals, with higher c_j given by universal curvature polynomials including boundary contributions. The authors develop a boundary-aware Wiener-Hopf factorization to capture edge effects, derive inclusion-exclusion principles, connective relations to Euler characteristic, and spectral-geometry analogues such as “hearing the shape of a drum” for magnitude; they also provide algorithmic tools and explicit examples (e.g., cylinders, shells, balls) illustrating the computation of c_j. This framework links magnitude to classical geometric invariants and residues, offering a computational pathway to higher-order curvature terms and enriching the interface between magnitude theory and spectral geometry. A Python-based algorithm implements the coefficient computation, enabling practical exploration of magnitude in varied geometric settings.
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.