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Construction of larger Riemannian metrics with bounded sectional curvatures and applications
Author(s) -
Chen Xin,
Wang FengYu
Publication year - 2008
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdn045
Subject(s) - mathematics , bounded function , sectional curvature , riemannian manifold , pure mathematics , minimal volume , conjecture , metric (unit) , riemannian submersion , fundamental theorem of riemannian geometry , riemannian geometry , exponential map (riemannian geometry) , manifold (fluid mechanics) , sobolev space , ricci curvature , information geometry , mathematical analysis , scalar curvature , curvature , geometry , mechanical engineering , operations management , economics , engineering
For any (not necessarily complete) Riemannian manifold, we construct a larger Riemannian metric which is complete and with bounded sectional curvatures. As an application, log‐Sobolev inequalities are established on arbitrary Riemannian manifolds with reference measures having smooth and strictly positive densities. In particular, a conjecture of the second‐named author (see [ J. Math. Anal. Appl. 300 (2004) 426–435]) is solved.