The magnitude and spectral geometry

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 R². 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 0 and admits a meromorphic continuation to sectors in C. We obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton: In the limit R→∞ 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.