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Global existence and convergence for a higher order flow in conformal geometry. (English) Zbl 1042.53016

The main purpose of this paper is to construct conformal metrics for which the curvature quantity \(Q\) is a constant multiple of a prescribed positive function \(f\) on a compact n-dimensional manifold \(M\). In section 2 the author derives the evolution equation for the conformal factor and the curvature quantity \(Q\), next, in section 3 he shows that the solution is bounded in \(H^{n/2}\). In sections 4 and 5 the author shows that the solution exists for all time, and in section 6 he proves that the evolution equation converges to a stationary solution. Finally, in section 8 he proves a compactness theorem for conformal metrics on \(S^n\) (proposition 1.4).

MSC:

53C20 Global Riemannian geometry, including pinching
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58D25 Equations in function spaces; evolution equations
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