Infinitely many solutions to the Yamabe problem on noncompact manifolds

Abstract

We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, S m × R d , m ≥ 2, d ≥ 1, and S m × H d , 2 ≤ d < m. As a consequence, we obtain infinitely many periodic solutions to the singular Yamabe problem on S m \ S k , for all 0 ≤ k < (m − 2)/2, the maximal range where nonuniqueness is possible. We also show that all Bieberbach groups in Iso(R d ) are periods of bifurcating branches of solutions to the Yamabe problem on S m × R d , m ≥ 2, d ≥ 1.

The first eigenvalue of a homogeneous CROSS

“…As an application of the characterization of stable homogeneous solutions to the Yamabe problem in the previous section, we now establish nonuniqueness results via Bifurcation Theory, along the lines of [BP13a,BP13b,BP18,dLPZ12]. Following these references, solutions to the Yamabe problem are said to bifurcate from a curve g(t) of solutions on M at t = t * if there exist a sequence of parameters t q converging to t * , and constant scalar curvature metrics g q ∈ [g(t q )] converging to g(t * ), such that Vol(M, g q ) = Vol(M, g(t q )) and g q = g(t q ), for all q ∈ N.…”

“…Combining the above classification of stable solutions to the Yamabe problem and classical results in Bifurcation Theory, it is possible to detect the existence of branches of solutions issuing from paths of homogeneous metrics when they lose stability, i.e., when (t 1 , t 2 , t 3 ) leaves the set S n . By uniqueness of homogeneous metrics in their conformal class, these bifurcating solutions must be inhomogeneous, fitting a wider context of symmetry-breaking bifurcations [BP13a,BP13b,BP18].…”

Subspace foliations and collapse of closed flat manifolds

“…Remark 5.4. The collapsing deformation of a flat manifold M π along a subspace foliation F W as formulated in (1.2) coincides with the notion of collapse of flat metrics from [BP18,BDP18]. Namely, the latter formulation is in terms of a deformation of the original Bieberbach group π ⊂ Aff(R n ) through (isomorphic) Bieberbach groups…”

“…Aside from its intrinsic geometric relevance, the existence of different collapsed limits of M π = R n /π enables one to construct different π-periodic solutions in R n to several geometric variational problems. For instance, this method was used to construct π-periodic solutions to the Yamabe problem on S m × R n in [BP18].…”

A variational analysis of the spinorial Yamabe equation on product manifolds

“…This problem goes back to the seminal work of Schoen [42] and has been deeply investigated in several works (see e.g. [17][18][19] and references therein). Together with Bettiol and Piccione [20], the first author investigated the multiplicity of constant Q−curvature metrics in a similar product manifold as in our setting but in the framework of Berger spheres.…”