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Rigidity Theorems for Diameter Estimates of Compact Manifold with Boundary

Li, Haizhong; Wei, Yong

Description

Let (N,g) be an n-dimensional complete Riemannian manifold with nonempty boundary ∂N. Assume that the Ricci curvature of (N,g) has a negative lower bound Ric≥−(n−1)c2 for some c>0, and the mean curvature of the boundary ∂N satisfies H≥(n−1)c0>(n−1)c for some c0>c>0. Then a known result (cf. [12]) says that supx∈Nd(x,∂N)≤1ccoth−1c0c⁠. In this paper, we prove the rigidity result that if N is compact, then the equality holds if and only if (N,g) is isometric to the geodesic ball of radius...[Show more]

dc.contributor.authorLi, Haizhong
dc.contributor.authorWei, Yong
dc.date.accessioned2019-11-25T23:40:07Z
dc.identifier.issn1073-7928
dc.identifier.urihttp://hdl.handle.net/1885/186628
dc.description.abstractLet (N,g) be an n-dimensional complete Riemannian manifold with nonempty boundary ∂N. Assume that the Ricci curvature of (N,g) has a negative lower bound Ric≥−(n−1)c2 for some c>0, and the mean curvature of the boundary ∂N satisfies H≥(n−1)c0>(n−1)c for some c0>c>0. Then a known result (cf. [12]) says that supx∈Nd(x,∂N)≤1ccoth−1c0c⁠. In this paper, we prove the rigidity result that if N is compact, then the equality holds if and only if (N,g) is isometric to the geodesic ball of radius 1ccoth−1c0c in an n-dimensional hyperbolic space Hn(−c2) of constant sectional curvature −c2. Moreover, we prove an analogous result for manifold with m-Bakry–Émery Ricci curvature bounded below by a negative constant.
dc.description.sponsorshipThe research of the authors was supported by NSFC No. 11271214.
dc.format.mimetypeapplication/pdf
dc.language.isoen_AU
dc.publisherDuke University Press
dc.rights© 2014 The Author(s)
dc.sourceInternational Mathematics Research Notices
dc.titleRigidity Theorems for Diameter Estimates of Compact Manifold with Boundary
dc.typeJournal article
local.description.notesImported from ARIES
local.identifier.citationvolume2015
dcterms.dateAccepted2014-03-10
dc.date.issued2014-04-02
local.identifier.absfor010102 - Algebraic and Differential Geometry
local.identifier.ariespublicationu5013521xPUB47
local.publisher.urlhttps://academic.oup.com
local.type.statusPublished Version
local.contributor.affiliationLi, Haizhong, Tsinghua University
local.contributor.affiliationWei, Yong, Tsinghua University
local.description.embargo2037-12-31
local.bibliographicCitation.issue11
local.bibliographicCitation.startpage3651
local.bibliographicCitation.lastpage3668
local.identifier.doi10.1093/imrn/rnu052
local.identifier.absseo970101 - Expanding Knowledge in the Mathematical Sciences
dc.date.updated2019-05-19T08:23:17Z
CollectionsANU Research Publications

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