Variational properties of the second eigenvalue of the conformal Laplacian

摘要

Let (M-n, g) be a closed Riemannian manifold of dimension n >= 3. Assume [g] is a conformal class for which the Conformal Laplacian L-g has at least two negative eigenvalues. We show the existence of a (generalized) metric that maximizes the second eigenvalue of L-g over all conformal metrics (the first eigenvalue is maximized by the Yamabe metric). We also show that a maximal metric defines either a nodal solution of the Yamabe equation, or a harmonic map to a sphere. Moreover, we construct examples of each possibility. (c) 2021 Elsevier Inc. All rights reserved.