2022/01/27 by Heiko Gimperlein, Gimperlein, Heiko, Magnus Goffeng +3
Mathematics · #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Metric Geometry (math.MG) #Spectral Theory (math.SP) #math.AP #math.CA #math.DG #math.MG #math.SP
paper · pdf · doi:10.48550/arxiv.2201.11363
published as American Journal of Mathematics 148 (2026), 1151-1198 · 33 pages, 5 figures, python code in ancillary file, to appear in American Journal of Mathematics
openalex publication_date 2022/01/27 · openalex created_date 2025/10/10 · arxiv created 2025/11/16 · openalex updated_date 2026/07/28 · arxiv updated 2026/08/04
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. In the semiclassical 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. We initiate the study of magnitude analogues to classical questions in spectral geometry and prove an asymptotic variant of the Leinster-Willerton conjecture.