The Equivariant Second Yamabe Constant

Abstract

For a closed Riemannian manifold of dimension n ≥ 3 and a subgroup G of the isometry group, we define and study the G−equivariant second Yamabe constant and we obtain some results on the existence of G−invariant nodal solutions of the Yamabe equation.

Isoparametric functions and nodal solutions of the Yamabe equation

“…Using similar arguments as the ones used in the proof of (Theorem 3.4, [2]) and (Theorem 1.5, [27]) it can be seen that if g u realizes the second isoparametric f −constant of (M × N, g + h) then u = |v 2 | and v 2 is a nodal solution of the Yamabe equation. Also, by the connectedness of M , it can be seen that the number of connected components of M × N − {v −1 2 (0)} is two.…”

“…For the details we refer to the reader to (Proposition 3.2 in [2]) and to Section 4 in [27]. The subspace V 0 := span(v 1 , v 2 ) realizes the generalized second eigenvalue λ f 2 (g u ).…”

“…Making use of this invariant they were able to prove existence results when the conformal class of the Riemannian manifold admits a Riemannian metric of non-negative scalar curvature. Further results on the existence of nodal solutions of the Yamabe equation using the variational approach of Ammann and Humbert can be found in Petean [44], El Sayed [17], Madani and the author [27]. We refer the reader to the articles by Clapp and Fernandez [9] and Fernandez and Petean [18] for recent results on the multiplicity of nodal solutions of the Yamabe equation.…”

Equivariant Yamabe problem with boundary

“…See also [11,14,15,19] for results related to the equivariant Yamabe problem. The Yamabe problem can be formulated for manifolds with boundary.…”

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

“…solutions that change sign). See for instance the articles [4,8,9,14,15,16,24] and the references in them. Nodal solutions u do not give metrics of constant scalar curvature since u vanishes at some points and therefore |u| pn−2 g is not a Riemannian metric.…”