[Paper Review] Conformal invariants from nodal sets

Yaiza Canzani, Rod Gover+2 more|arXiv (Cornell University)|Aug 15, 2012

Pelvic and Acetabular Injuries7 citations

TL;DR

This paper introduces new conformal invariants derived from nodal sets and negative eigenvalues of GJMS operators—such as the Yamabe and Paneitz operators—on Riemannian manifolds of dimension $ n \geq 3 $. It establishes a conformal version of Courant's Nodal Domain Theorem, proves the Yamabe operator can have arbitrarily many negative eigenvalues, and applies these invariants to curvature prescription problems, particularly for $ Q $-curvature.

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.

Motivation & Objective

• To define and study conformal invariants arising from nodal sets and negative eigenvalues of conformally covariant operators, particularly GJMS operators.
• To extend Courant's Nodal Domain Theorem to the conformal geometry setting.
• To investigate the existence of metrics on $ n $-dimensional manifolds ($ n \geq 3 $) for which these invariants are nontrivial.
• To analyze the spectral properties of the Yamabe and Paneitz operators, especially the number of negative eigenvalues.
• To address curvature prescription problems, including $ Q $-curvature prescription for non-critical $ Q $-curvatures via the appendix study.

Proposed method

• Utilizes the GJMS operators, which generalize the Yamabe and Paneitz operators, as conformally covariant differential operators on Riemannian manifolds.
• Analyzes nodal sets of eigenfunctions of these operators to extract conformal invariants under conformal changes of metric.
• Applies spectral theory to show that the Yamabe operator can possess an arbitrarily large number of negative eigenvalues on any $ n $-manifold with $ n \geq 3 $.
• Employs variational and perturbative techniques to construct metrics with nontrivial invariants, particularly on Einstein and Heisenberg manifolds.
• Uses the framework of conformal geometry to relate nodal set structure to spectral invariants and curvature prescription.
• In the appendix, applies variational methods to $ Q $-curvature prescription problems for non-critical $ Q $-curvatures, building on the main results.

Experimental results

Research questions

• RQ1 Can conformal invariants be constructed from nodal sets of eigenfunctions of GJMS operators?
• RQ2 To what extent can the number of negative eigenvalues of the Yamabe operator be controlled or made arbitrarily large?
• RQ3 How do nodal set structures relate to conformal invariants in the context of curvature prescription?
• RQ4 What is the behavior of these invariants on specific geometric spaces such as Heisenberg manifolds?
• RQ5 Can the $ Q $-curvature prescription problem be solved for non-critical $ Q $-curvatures using the spectral and geometric tools developed?

Key findings

• On any Riemannian manifold of dimension $ n \geq 3 $, there exist metrics for which the conformal invariants derived from nodal sets and negative eigenvalues of the Yamabe operator are nontrivial.
• The Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension $ n \geq 3 $, demonstrating high spectral complexity.
• A conformal version of Courant's Nodal Domain Theorem is established, linking nodal domain counts to conformal invariants.
• For certain Einstein and Heisenberg manifolds, higher-order GJMS operators also exhibit arbitrarily large numbers of negative eigenvalues.
• The invariants associated with the Yamabe and Paneitz operators on left-invariant metrics of Heisenberg manifolds are explicitly described.
• The appendix provides a solution to the $ Q $-curvature prescription problem for non-critical $ Q $-curvatures using variational methods.