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- Title
Prescribed Q-curvature flow on closed manifolds of even dimension.
- Authors
Ngô, Quốc Anh; Zhang, Hong
- Abstract
On a closed Riemannian manifold (M , g 0) of even dimension n ≥ 4 , the well-known prescribed Q-curvature problem asks whether there is a metric g comformal to g 0 such that its Q-curvature, associated with the GJMS operator P g , is equal to a given function f. Letting g = e 2 u g 0 , this problem is equivalent to solving P g 0 u + Q g 0 = f e nu , where Q g 0 denotes the Q-curvature of g 0 . The primary objective of the paper is to introduce the following negative gradient flow of the time dependent metric g(t) conformal to g 0 , ∂ g (t) ∂ t = - 2 Q g (t) - ∫ M f Q g (t) d μ g (t) ∫ M f 2 d μ g (t) f g (t) for t > 0 , to study the problem of prescribing Q-curvature. Since ∫ M Q g d μ g is conformally invariant, our analysis depends on the size of ∫ M Q g 0 d μ g 0 which is assumed to satisfy ∫ M Q 0 d μ g 0 ≠ k (n - 1) ! vol (S n) for all k = 2 , 3 , ... The paper is twofold. First, we identify suitable conditions on f such that the gradient flow defined as above is defined to all time and convergent, as time goes to infinity, sequentially or uniformly. Second, we show that various existence results for prescribed Q-curvature problem can be derived from the convergence of the flow.
- Subjects
MANIFOLDS (Mathematics); RIEMANNIAN manifolds; INFINITY (Mathematics); CONFORMAL geometry
- Publication
Calculus of Variations & Partial Differential Equations, 2020, Vol 59, Issue 4, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-020-01780-y