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Q Curvature on a Class of 3‐Manifolds
Author(s) -
Hang Fengbo,
Yang Paul C.
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21559
Subject(s) - mathematics , scalar curvature , conformal map , curvature , pure mathematics , prescribed scalar curvature problem , mathematical analysis , yamabe flow , riemannian manifold , sectional curvature , constant (computer programming) , compact space , manifold (fluid mechanics) , geometry , computer science , mechanical engineering , engineering , programming language
Motivated by the strong maximum principle for the Paneitz operator in dimension 5 or higher found in a preprint by Gursky and Malchiodi and the calculation of the second variation of the Green's function pole's value on 3 in our preprint, we study the Riemannian metric on 3‐manifolds with positive scalar and Q curvature. Among other things, we show it is always possible to find a constant Q curvature metric in the conformal class. Moreover, the Green's function is always negative away from the pole, and the pole's value vanishes if and only if the Riemannian manifold is conformal diffeomorphic to the standard 3 . Compactness of constant Q curvature metrics in a conformal class and the associated Sobolev inequality are also discussed. © 2016 Wiley Periodicals, Inc.