The second Yamabe invariant

Abstract

Let (M, g) be a compact Riemannian manifold of dimension n 3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.

Isoparametric functions and nodal solutions of the Yamabe equation

“…Let (M, g) be a closed connected Riemannian manifold. A smooth non-constant function f : M −→ R is called an isoparametric function is there exist a smooth function b and a continuous function a such that (2)…”

“…When t is a regular value of f we say that the level set M t is an isoparametric hypersurface. The condition (2) implies that the regular level sets of f are parallel hypersurfaces. On the other hand, by condition (3) under condition (2), the level sets of f have constant mean curvature.…”

“…Among the authors that have studied nodal solutions of semi-linear elliptic equations on Riemannian manifolds we can mention Ding [13], Hebey and Vaugon [24], Holcman [28], Jourdain [31], Djadli and Jourdain [15], and del Pino et al [14]. But the first general result on the existence of nodal solution of Yamabe equation was obtained by Ammann and Humbert in [2]. They introduced the so called second Yamabe constant which is a conformal invariant induced by the second eigenvalue of the conformal Laplacian operator (see Section 2.2, Preliminaries).…”

A note on solutions of Yamabe-type equations on products of spheres

“…But they have geometric interest. The existence of at least one nodal solution is proved in general cases in [4], as minimizers for the second Yamabe invariant. But there are not as many results about multiplicity of nodal solutions as in the positive case.…”

The Equivariant Second Yamabe Constant

“…We say that u is a solution of the Yamabe equation if for some constant c, the function u satisfies (2) L g (u) = c|u| pn−2 u, where L g := a n ∆ g + s g is the conformal Laplacian of (M, g), a n := 4(n−1) n−2 , and p n := 2n n−2 . A nodal solution of the Yamabe equation of (M, g) is a solution of (2) that changes sign.…”