Monopole metrics and the orbifold Yamabe problem

TL;DR

This paper studies the behavior of constant scalar curvature metrics in certain self-dual conformal classes on connected sums of complex projective planes, revealing limitations in solving the orbifold Yamabe problem.

Contribution

It analyzes the limiting behavior of these metrics as monopole points approach each other or the boundary, and demonstrates non-solvability of the orbifold Yamabe problem in specific cases.

Findings

No constant scalar curvature orbifold metric exists in the conformal class of certain hyperkähler ALE spaces.

The orbifold Yamabe problem is not always solvable for these conformal classes.

Limiting behaviors of metrics depend on the configuration of monopole points.

Abstract

We consider the self-dual conformal classes on n#CP^2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3-space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the conformal class of a conformally compactified non-flat hyperkahler ALE space in dimension four.