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

By M. Gursky, A. Malchiodi. 2014

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In this paper we consider Riemannian manifolds (M, 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/4c8b418315f786bebfe36110c23acc83d26fe7a1

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In Applied Mathematics

Citations in the corpus

Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities (2019, background): “See also Gursky-Malchoidi [22] and Hang-Yang [24] for the recent progresses on the fourth order Q-curvature problem on Riemannian manifolds, where the positivity of the scalar curvature and the positivity of the Yamabe invariant are assumed, respectively.”
Geometric inequalities for manifolds with Ricci curvature in the Kato class (2016, background): “Machioldi ([35]) have discovered some new maximum principles for the Paneitz operator when the Q-curvature is nonnegative and the scalar curvature is positive.”
The Nirenberg problem and its generalizations: a unified approach (2014, background): “[58], Gursky and Malchiodi [41], Hang and Yang [44–46] and the references therein.”