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arXiv · 1401.3216

A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$-curvature

Matthew J. Gursky·Andrea Malchiodi
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Abstract

In this paper we consider Riemannian manifolds $(M^n,g)$ of dimension $n \geq 5$, with semi-positive $Q$-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 $Q$-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 $Q$-curvature.

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Matthew J. Gursky, Andrea Malchiodi. 2014-01-14. A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$-curvature. https://arxiv.org/abs/1401.3216

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