%0 Journal Article
%D 2014
%T Conformal invariants from nodal sets. I. negative eigenvalues and curvature prescription
%A Rod R. Gover
%A Yaiza Canzani
%A Dmitry Jakobson
%A Raphaël Ponge
%A Andrea Malchiodi
%X 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'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.
%I Oxford University Press
%G en
%U http://urania.sissa.it/xmlui/handle/1963/35128
%1 35366
%2 Mathematics
%4 1
%$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-12-02T16:09:57Z
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%R 10.1093/imrn/rns295