Blow-up Solutions for Linear Perturbations of the Yamabe Equation

Abstract

Abstract. For a smooth, compact Riemannian manifold (M, g) of dimension N ≥ 3, we are interested in the critical equationwhere ∆ g is the Laplace-Beltrami operator, S g is the Scalar curvature of (M, g), h ∈ C 0,α (M ), and ε is a small parameter.

Blowing-up solutions for the Yamabe equation

“…Remark 2.1. In [8] a similar construction is performed. Thanks to some symmetries properties, it is shown there that g ¼ 0 and the function V can be reduced to a simpler expression.…”

“…where Z i m ðxÞ :¼ m ÀðNÀ2Þ=2 Z i ðx=mÞ are defined in (7) and (8). We then define the projections P m; x and P ?…”

“…This is done in Section 2. A similar idea has been already used by Esposito-Pistoia-Ve ´tois in [7], [8]. In Section 3, we reduce the problem to a finite dimensional one, we study the critical points of the corresponding finite dimensional functional, i.e.…”

The effect of linear perturbations on the Yamabe problem

“…An important point here is that we allow the metric g to vary in the conformal class so to gain flatness at each point ξ ∈ M. An alternative, less geometric approach can be devised in the non-l.c.f. case, see Esposito-Pistoia-Vétois [20], by keeping g fixed and slightly correcting the basic ansatz via linearization so to account for the local geometry. Thanks to the solvability theory of the linearized operator for (1.3) at W ε,t,ξ , we are led to study critical points of a finite-dimensional functional J ε (t, ξ).…”