Moduli Spaces of Singular Yamabe Metrics

TL;DR

This paper investigates the structure of the moduli space of singular Yamabe metrics on the sphere, establishing its local analytic variety structure and dimension, through advanced linear and nonlinear analysis techniques.

Contribution

It develops a Fredholm and asymptotic regularity theory for these metrics, showing the moduli space is a real analytic variety of dimension k and a manifold for generic cases.

Findings

The moduli space is a locally real analytic variety of dimension k.

For generic conformal classes, the moduli space is a k-dimensional real analytic manifold.

The paper introduces a Fredholm theory for asymptotically periodic manifolds.

Abstract

Complete, conformally flat metrics of constant positive scalar curvature on the complement of $k$ points in the $n$-sphere, $k \ge 2$, $n \ge 3$, were constructed by R\. Schoen [S2]. 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 $k$. For a generic set of nearby conformal classes the moduli space is shown to be a $k-$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.