Conformal invariants from nodal sets. I: Negative eigenvalues and curvature prescription. With an appendix by Rod Gover and Andrea Malchiodi (Q2929680)

In this paper conformal invariants of conformally covariant differential operators are investigated. In particular, the Graham, Jenne, Mason and Sparling (GJMS) operators are considered. These are generalizations of the Yamabe and the Paneitz operator. It is shown that nodal sets and nodal domains of any null-eigenfunction are conformal invariants. Another conformal invariant is the number of negative eigenvalues. It is shown that the number of negative eigenvalues of the Yamabe operator can be arbitrarily large. The following version of Courant's nodal domain theorem is shown: If the Yamabe operator has \(m\) negative eigenvalues then its null-eigenfunctions have at most \(m+1\) nodal domains.NEWLINENEWLINEFor Part II, see [\textit{G. Cox} et al., J. Spectr. Theory 11, No. 2, 387--409 (2021; Zbl 1477.58018)].