%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 No. of bitstreams: 1 preprint2014.pdf: 356671 bytes, checksum: 20e817f9f20d9c72d717e04f94f86bd9 (MD5) %R 10.1093/imrn/rns295