The magnitude and spectral geometry
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912710678020096 |
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| author | Gimperlein, Heiko Goffeng, Magnus Louca, Nikoletta |
| author_facet | Gimperlein, Heiko Goffeng, Magnus Louca, Nikoletta |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_11363 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The magnitude and spectral geometry Gimperlein, Heiko Goffeng, Magnus Louca, Nikoletta Differential Geometry Analysis of PDEs Classical Analysis and ODEs Metric Geometry Spectral Theory 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. |
| title | The magnitude and spectral geometry |
| topic | Differential Geometry Analysis of PDEs Classical Analysis and ODEs Metric Geometry Spectral Theory |
| url | https://arxiv.org/abs/2201.11363 |