Vol. 302, No. 1, 2019

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A compactness theorem on Branson's $Q$-curvature equation

Gang Li

Vol. 302 (2019), No. 1, 119–179
Abstract

Let (M,g) be a closed Riemannian manifold of dimension 5 n 7. Assume that (M,g) is not conformally equivalent to the round sphere. If the scalar curvature Rg is greater than or equal to 0 and the Q-curvature Qg is greater than or equal to 0 on M with Qg(p) > 0 for some point p M, we prove that the set of metrics in the conformal class of g with prescribed constant positive Q-curvature is compact in C4,α for any 0 < α < 1.

Keywords
compactness, constant $Q$-curvature metrics, blowup argument, positive mass theorem, maximum principle, fourth order elliptic equations
Mathematical Subject Classification 2010
Primary: 53C21
Secondary: 35B50, 35J61, 53A30
Milestones
Received: 21 December 2017
Revised: 4 January 2019
Accepted: 13 March 2019
Published: 5 November 2019
Authors
Gang Li
Beijing International Center for Mathematical Research
Peking University
Beijing
China
Department of Mathematics
Shandong University
Ji’an
China