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Moduli Spaces of Singular Yamabe Metrics

Authors: Rafe Mazzeo, Daniel Pollack and Karen Uhlenbeck
Journal: J. Amer. Math. Soc. 9 (1996), 303-344
MSC (1991): Primary 58D27
MathSciNet review: 1356375
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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 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 $k$. 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.

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Additional Information

Rafe Mazzeo
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Daniel Pollack
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Karen Uhlenbeck
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712

Received by editor(s): January 20, 1994
Additional Notes: The first author’s research was supported in part by NSF Young Investigator Award, the Sloan Foundation, NSF grant # DMS9001702, the second author’s research was supported by NSF grant # DMS9022140, and the third author’s research was supported by the Sid Richardson and O’Donnell foundations.
Article copyright: © Copyright 1996 American Mathematical Society