@article {2014,
	title = {Conformal invariants from nodal sets. I. negative eigenvalues and curvature prescription},
	number = {International Mathematics Research Notices;volume 2014; issue 9; pages 2356-2400;},
	year = {2014},
	publisher = {Oxford University Press},
	abstract = {In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the Graham, Jenne, Mason, and Sparling (GJMS) operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant{\textquoteright}s Nodal Domain Theorem. We also show that on any manifold of dimension n>=3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n>=3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. We describe the invariants arising from the Yamabe and Paneitz operators associated to left-invariant metrics on Heisenberg manifolds. Finally, in Appendix, the second named author and Andrea Malchiodi study the Q-curvature prescription problems for noncritical Q-curvatures.},
	doi = {10.1093/imrn/rns295},
	url = {http://urania.sissa.it/xmlui/handle/1963/35128},
	author = {Rod R. Gover and Yaiza Canzani and Dmitry Jakobson and Rapha{\"e}l Ponge and Andrea Malchiodi}
}
