Volume growth on manifolds with more than one end
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909180815736832 |
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| author | Das, Anushree Maity, Soma |
| author_facet | Das, Anushree Maity, Soma |
| contents | For an open manifold $M$ and a function $v$ with bounded growth of derivative, there exists a Riemannian metric of bounded geometry on $M$ such that the volume growth function lies in the same growth class as $v$. This was proved by R. Grimaldi and P. Pansu with the proof focusing on the case of manifolds with a single end. We prove this in the case of manifolds with multiple ends and call the constructed metrics Grimaldi-Pansu metrics. We give uniform bounds for the volume growth function of these metrics in terms of the given bgd-function in the case of a certain class of manifolds which can be written as connected sums of a finite collection of closed and compact manifolds. We study the volume doubling condition and the Relatively Connected Annulus (R.C.A.) property of the Grimaldi-Pansu metrics, which play an important role in studying geometric analysis on manifolds with finitely many ends. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06868 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Volume growth on manifolds with more than one end Das, Anushree Maity, Soma Differential Geometry Geometric Topology 51F30, 53C21, 53C23 For an open manifold $M$ and a function $v$ with bounded growth of derivative, there exists a Riemannian metric of bounded geometry on $M$ such that the volume growth function lies in the same growth class as $v$. This was proved by R. Grimaldi and P. Pansu with the proof focusing on the case of manifolds with a single end. We prove this in the case of manifolds with multiple ends and call the constructed metrics Grimaldi-Pansu metrics. We give uniform bounds for the volume growth function of these metrics in terms of the given bgd-function in the case of a certain class of manifolds which can be written as connected sums of a finite collection of closed and compact manifolds. We study the volume doubling condition and the Relatively Connected Annulus (R.C.A.) property of the Grimaldi-Pansu metrics, which play an important role in studying geometric analysis on manifolds with finitely many ends. |
| title | Volume growth on manifolds with more than one end |
| topic | Differential Geometry Geometric Topology 51F30, 53C21, 53C23 |
| url | https://arxiv.org/abs/2309.06868 |