TY - JOUR
T1 - Conformal invariants from nodal sets. I. negative eigenvalues and curvature prescription
Y1 - 2014
A1 - Rod R. Gover
A1 - Yaiza Canzani
A1 - Dmitry Jakobson
A1 - Raphaël Ponge
A1 - Andrea Malchiodi
AB - 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.
PB - Oxford University Press
UR - http://urania.sissa.it/xmlui/handle/1963/35128
U1 - 35366
U2 - Mathematics
U4 - 1
ER -