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

Matthew J. Gursky; Andrea Malchiodi

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 9, page 2137-2173
- ISSN: 1435-9855

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topGursky, 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 -

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