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A Monge-Ampère equation in conformal geometry

Matthew J. Gursky — 2008

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider the Monge-Ampère-type equation det ( A + λ g ) = const . , where A is the Schouten tensor of a conformally related metric and λ > 0 is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique.

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

Matthew J. GurskyAndrea Malchiodi — 2015

Journal of the European Mathematical Society

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...

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