Isoperimetric Clusters on Riemannian and Finsler Manifolds: Properties and Applica...
Geometry of manifolds in the euclidian space and in the Minkowski space
Geometry of Riemannian, semi-Riemannian varieties and actions of Lie groups
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| Author(s): |
Total Authors: 2
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| Affiliation: | [1] Univ Fed ABC, Ctr Matemat Comp & Cognicao, Ave Estados 5001, BR-09210580 Santo Andre, SP - Brazil
[2] Univ Sao Paulo, Inst Matemat & Estat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil
Total Affiliations: 2
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| Document type: | Journal article |
| Source: | INTERNATIONAL MATHEMATICS RESEARCH NOTICES; v. 2020, n. 15, p. 4667-4720, AUG 2020. |
| Web of Science Citations: | 1 |
| Abstract | |
We provide an isoperimetric comparison theorem for small volumes in an n-dimensional Riemannian manifold (M-n, g) with C-3 bounded geometry in a suitable sense involving the scalar curvature function. Under C-3 bounds of the geometry, if the supremum of scalar curvature function S-g < n(n - 1)k(0) for some k(0) is an element of R, then for small volumes the isoperimetric profile of (M-n, g) is less then or equal to the isoperimetric profile of the complete simply connected space form of constant sectional curvature k(0). This work generalizes Theorem 2 of {[}12] in which the same result was proved in the case where (M-n, g) is assumed to be compact. As a consequence of our result we give an asymptotic expansion in Puiseux series up to the 2nd nontrivial term of the isoperimetric profile function for small volumes, generalizing our earlier asymptotic expansion {[}29]. Finally, as a corollary of our isoperimetric comparison result, it is shown that for small volumes the Aubin-Cartan-Hadamard's conjecture is true in any dimension n in the special case of manifolds with C-3 bounded geometry, and S-g < n(n - 1)k(0). Two different intrinsic proofs of the fact that an isoperimetric region of small volume is of small diameter. The 1st under the assumption of mild bounded geometry, that is, positive injectivity radius and Ricci curvature bounded below. The 2nd assuming the existence of an upper bound of the sectional curvature, positive injectivity radius, and a lower bound of the Ricci curvature. (C) The Author(s) 2018. Published by Oxford University Press. All rights reserved. (AU) | |
| FAPESP's process: | 17/13155-3 - Geometric measure Theory and Isoperimetric problems |
| Grantee: | Luis Eduardo Osorio Acevedo |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |