In this paper we consider Riemannian manifolds of dimension , with semi-positive -curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green's function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive -curvature. Modifying the test function construction of Esposito-Robert, we show that it is possible to choose an initial conformal metric so that the flow has a sequential limit which is smooth and positive, and defines a conformal metric of constant positive -curvature.
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Matthew J. Gursky, Andrea Malchiodi, A strong maximum principle for the Paneitz operator and a non-local flow for the -curvature. J. Eur. Math. Soc. 17 (2015), no. 9, pp. 2137–2173DOI 10.4171/JEMS/553