Conformal invariants from nodal sets. I. Negative Eigenvalues and Curvature Prescription

TL;DR

This paper investigates conformal invariants derived from nodal sets and negative eigenvalues of GJMS operators, applying these to curvature prescription problems and demonstrating the existence of metrics with nontrivial invariants.

Contribution

It introduces new conformal invariants from nodal sets and negative eigenvalues, and applies these to curvature prescription problems, including results on the Yamabe and Paneitz operators.

Findings

Existence of metrics with nontrivial invariants on manifolds of dimension ≥ 3

Yamabe operator can have arbitrarily many negative eigenvalues

Results for higher order GJMS operators on Einstein and Heisenberg manifolds

Abstract

In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the 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 $n\geq 3$, 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 $n\geq 3$. 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 the appendix, the 2nd named author and Andrea Malchiodi study the $Q$-curvature prescription problems for non-critical $Q$-curvatures.