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

Gursky, Matthew J., and Malchiodi, Andrea. "A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$-curvature." Journal of the European Mathematical Society 017.9 (2015): 2137-2173. <http://eudml.org/doc/277165>.

@article{Gursky2015,
abstract = {In this paper we consider Riemannian manifolds $(M^n,g)$ of dimension $n \ge 5$, 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.},
author = {Gursky, Matthew J., Malchiodi, Andrea},
journal = {Journal of the European Mathematical Society},
keywords = {$Q$-curvature; Paneitz operator; conformal geometry; non-local flow; eponymous operator of Paneitz; conformal geometry; eponymous operator of Paneitz},
language = {eng},
number = {9},
pages = {2137-2173},
publisher = {European Mathematical Society Publishing House},
title = {A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$-curvature},
url = {http://eudml.org/doc/277165},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Gursky, Matthew J.
AU - Malchiodi, Andrea
TI - A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$-curvature
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 9
SP - 2137
EP - 2173
AB - In this paper we consider Riemannian manifolds $(M^n,g)$ of dimension $n \ge 5$, 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.
LA - eng
KW - $Q$-curvature; Paneitz operator; conformal geometry; non-local flow; eponymous operator of Paneitz; conformal geometry; eponymous operator of Paneitz
UR - http://eudml.org/doc/277165
ER -