The magnitude and spectral geometry

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 $\mathbb{C}$. We obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton: In the 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. As a consequence, we initiate the study of magnitude analogues to classical questions in spectral geometry.