Riesz energies and the magnitude of manifolds
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- by Heiko Gimperlein and Magnus Goffeng;
- Proc. Amer. Math. Soc.
- DOI: https://doi.org/10.1090/proc/17203
- Published electronically: April 14, 2025
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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.References
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Bibliographic Information
- Heiko Gimperlein
- Affiliation: Engineering Mathematics, Universität Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria
- MR Author ID: 922641
- ORCID: 0000-0003-3145-3021
- Email: heiko.gimperlein@uibk.ac.at
- Magnus Goffeng
- Affiliation: Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden
- MR Author ID: 895436
- ORCID: 0000-0002-4411-5104
- Email: magnus.goffeng@math.lth.se
- Received by editor(s): October 2, 2024
- Received by editor(s) in revised form: December 24, 2024
- Published electronically: April 14, 2025
- Additional Notes: The second listed author was supported by the Swedish Research Council Grant VR 2018-0350.
- Communicated by: Tanya Christiansen
- © Copyright 2025 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
- MSC (2020): Primary 58J50, 58J40; Secondary 51F99, 52A39, 31B15
- DOI: https://doi.org/10.1090/proc/17203