In this paper we consider Riemannian manifolds (M-n, g) of dimension n >= 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.

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

MALCHIODI, ANDREA
2015

Abstract

In this paper we consider Riemannian manifolds (M-n, g) of dimension n >= 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.
Settore MAT/05 - Analisi Matematica
Q-curvature, Paneitz operator, conformal geometry, non-local flow
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11384/60616
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