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Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities
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Scharrer, Christian (2022) Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities. Annals of Global Analysis and Geometry . doi:10.1007/s10455-021-09822-0 ISSN 0232-704X.
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WRAP-some-geometric-inequalities-varifolds-Riemannian-manifolds-based-monotonicity-identities-Scharrer-2022.pdf - Published Version - Requires a PDF viewer. Available under License Creative Commons Attribution 4.0. Download (509Kb) | Preview |
Official URL: http://dx.doi.org/10.1007/s10455-021-09822-0
Abstract
Using Rauch’s comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li–Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon-type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch’s comparison theorem, the proofs mainly rely on the first variation formula and thus are valid for general varifolds.
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Varifolds, Riemannian manifolds, Diameter (Geometry), Submanifolds, Sobolev spaces, Curvature, Geometric measure theory | ||||||
| Journal or Publication Title: | Annals of Global Analysis and Geometry | ||||||
| Publisher: | Springer | ||||||
| ISSN: | 0232-704X | ||||||
| Official Date: | 28 January 2022 | ||||||
| Dates: |
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| DOI: | 10.1007/s10455-021-09822-0 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||
| Date of first compliant deposit: | 22 February 2022 | ||||||
| Date of first compliant Open Access: | 24 February 2022 | ||||||
| RIOXX Funder/Project Grant: |
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