Conformal Invariants from Nodal Sets. I. Negative Eigenvalues and Curvature Prescription

Abstract

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 ≥ 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 the appendix, the 2nd named author and Andrea Malchiodi study the Q-curvature prescription problems for non-critical Q-curvatures.

Conformally covariant operators and conformal invariants on weighted graphs

“…As the multiplicity of zero eigenvalues of a matrix is the dimension of its nullspace, notice that Proof. The proof is similar to the proof of the corresponding result in [11,12]. Assume for contradiction that there exist two weights w 1 ∈ [w 0 ] such that the signatures of S w1 and S w0 are different.…”

“…The conformal class [w 0 ] is a path connected space, hence there exists a curve w t , t ∈ [0, 1] starting at w 0 and ending at w 1 . The 3 The manner in which these differential operators transform under a conformal change of weight is analogous to how the GMJS operators transform under a conformal change of Riemannian metric in [11,12]. eigenvalues of S wt depend continuously on t, and the multiplicity of 0 is constant by Corollary 4.4.…”

“…It follows easily that ker P g1 = e −bω ker P g0 . Based on this observation, the authors of the papers [11,12] constructed several conformal invariants related to the nodal sets of eigenfunctions in ker P g (that change sign). In the current paper, the authors initiate the development of the theory of conformally covariant operators on graphs, giving several examples of such operators and providing discrete counterparts to several results in [11,12].…”

“…We provide examples of conformally covariant operators, which include the edge Laplacian and the adjacency matrix on graphs. In the case where such an operator has a nontrivial kernel, we construct conformal invariants, providing discrete counterparts of several results in [11,12] established for Riemannian manifolds. In particular, we show that the nodal sets and nodal domains of null eigenvectors are conformal invariants.…”

On the prescribed Q-curvature problem in Riemannian manifolds

“…As the case treated here, these operators, the so-called of GJMS operators, have associated curvature invariants Q k,g . For more detail, see [18], [7], [32] and [33]. Furthermore, it was proved in [4] that given a closed locally conformally flat manifold (M, g) of even dimension n, we have that C nˆM Q k,g dV = χ(M), where C n = 1 ((n−2)!!…”