JournalsjemsVol. 17, No. 9pp. 2137–2173

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

  • Matthew J. Gursky

    University of Notre Dame, United States
  • Andrea Malchiodi

    Scuola Normale Superiore, Pisa, Italy
A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$-curvature cover
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Abstract

In this paper we consider Riemannian manifolds (Mn,g)(M^n,g) of dimension n5n \geq 5, with semi-positive QQ-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 QQ-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 QQ-curvature.

Cite this article

Matthew J. Gursky, Andrea Malchiodi, A strong maximum principle for the Paneitz operator and a non-local flow for the QQ-curvature. J. Eur. Math. Soc. 17 (2015), no. 9, pp. 2137–2173

DOI 10.4171/JEMS/553