On bifurcation of solutions of the Yamabe problem in product manifolds

Abstract

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non-homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds

Harmonically free group actions and equivariant bifurcation in the Yamabe problem

“…there is a sequence of solutions g sn of the Yamabe problem converging to g s * , each of which represents a second solution of the Yamabe problem in the correspondent g s -normalized conformal class. These were the same results obtained by Lima, Piccione and Zedda in [20], when studying multiplicity of solutions of the Yamabe problem in product manifolds (without boundary).…”

“…By hypothesis, π − a e π − b are non-equivalents. Under that conditions, [20,Theorem A.3] ensures the existence of a bifurcation instant s * ∈ (a, b) for the family of solutions, s → φ s ∈ D 0 s , of the equation…”

“…We showed that there is a countable subset S ⊂ (0, ∞) of instants such that, for all instants s ∈ (0, ∞)\S, the family {g s } s>0 is rigid, that is, save homotheties, there exists local uniqueness of solution of the Yamabe problem in the normalized conformal classes of the corresponding metrics g s . On the other hand, proposing a natural extension of bifurcation theorem of [20], we proved that, except for a finite number of instants, all the others s * ∈ S are bifurcation instants, i.e. there is a sequence of solutions g sn of the Yamabe problem converging to g s * , each of which represents a second solution of the Yamabe problem in the correspondent g s -normalized conformal class.…”

“…In 2012, Lima, Piccione and Zedda (in [20]) proposed a slightly more general statement of an abstract bifurcation result of Smoller and Wasserman [26] to determine the existence of multiple constant scalar curvature metrics on products of compact manifolds without boundary, based on a jump in the Morse index in a variational context. In fact, bifurcation techniques in the Yamabe problem have been used in many problems like in the case of collapsing Riemannian manifolds, and in the case of homogeneous metrics on spheres, see references [7] e [8].…”

“…In particular, in Subsection 2.3 we prove some simple, but important results about isometric actions and representations in our problem context. In Section 3, we remember the rigidity result of [19], naturally extended for manifolds with boundary in [10] and present a natural extension of the equivariant bifurcation theorem ( [20,Theorem 3.4]) to the case of manifolds with boundary (Theorem 3.7). In Section 4 we discuss the harmonically free action concept and, finally, in Section 5 we present our main result, Theorem 5.1.…”