Moduli Spaces of Singular Yamabe Metrics

Abstract

Complete, conformally flat metrics of constant positive scalar curvature on the complement of k k points in the n n -sphere, k ≥ 2 k \ge 2 , n ≥ 3 n \ge 3 , were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension k k . For a generic set of nearby conformal classes the moduli space is shown to be a k k -dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds.

The Yamabe problem on subdomains of even-dimensional spheres

“…Later on, R. Mazzeo and N. Smale [7] have proved that the answer is positive in the case where Λ is a perturbation of a k-dimensional sphere, for any 0 < k < (n − 2)/2. More recently, R. Mazzeo, D. Pollack and K. Uhlenbeck [5] have studied the moduli space of solutions to this problem when Λ is a finite set of points. The technique they have developed also allowed them to build solutions when Λ consists of a finite number of dipoles (pairs of points).…”

“…Using the conformal invariance of our problem, we may always perform some rotation and assume that the north pole of S n does not belong to Λ. After the stereographic projection from the north pole, our problem reduces to the resolution in R n \ Λ of the equation 9−∆ δ n ij u = u (n+2)/(n−2) (see for example (4) and (5)). Here and in the sequel, Λ will denote the stereographic projection of the set Λ.…”

Moduli space theory for the Allen-Cahn equation in the plane

“…The proofs of our results follow from the application of the implicit function theorem in a suitably designed weighted function space. The results and the arguments are very much in the spirit of what has already been done in the study of the moduli spaces of complete noncompact constant mean curvatures surfaces in Euclidean space or complete constant scalar curvature metrics in conformal geometry; see [15], [18] and [17].…”

On the Negativity of Ricci Curvatures of Complete Conformal Metrics

“…For the case of positive scalar curvatures, the existence and asymptotic behaviors of solutions have been extensively studied over the years. We shall not discuss this case here, but refer to [5,12,[19][20][21]23].…”