The Prescribed $Q$-Curvature Flow for Arbitrary Even Dimension in a Critical Case

In this paper, we study the prescribed Q-curvature flow equation on a arbitrary even dimensional closed Riemannian manifold (M, g), which was introduced by S. Brendle in [3], where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and ∫ M Qdμ < (n − 1)!Vol (S ). In this paper we study the critical case that ∫ M Qdμ = (n − 1)!Vol (S ), we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Liu’s existence result in [25] in dimensiona 4 and extend the work of Li-Zhu [26] in dimension 2 to general even dimensions. In the proof, we give a explicit expression of the limit of the corresponding energy functional when the blow up occurs.

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