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Dissertation/Thesis Abstract

Nonuniqueness of constant scalar curvature metrics in a conformal class
by Cohn, Zachary, Ph.D., Stanford University, 2009, 56; 3364497
Abstract (Summary)

The Yamabe Problem asks when the conformal class of a compact, Riemannian manifold (M, g) contains a metric of constant scalar curvature. Its resolution established that it always does, and that the sign of this constant is a conformal invariant.

We consider metrics whose conformal class includes a metric of constant positive scalar curvature. We show that given a smooth manifold ( M, g) of dimension n ≥ 9, there exists a metric which is arbitrarily close to g in the C1,α topology and whose conformal class contains an arbitrary number of distinct metrics with constant scalar curvature equal to 1. If we assume, in addition, that (M, g) is locally conformally flat, we may take to be close to g in the Cs topology for any s < [special characters omitted]</p>

These results generalize, in dimensions n ≥ 9, earlier results of Ambrosetti, Ambrosetti and Malchiodi, Berti and Malchiodi, and Pollack. Our proof constructs parameterized perturbations of an explicit approximate solution. The conformal class containing the constant scalar curvature metrics is obtained in this manner, and so has a well-understood geometry.

Indexing (document details)
Advisor: Schoen, Richard
School: Stanford University
School Location: United States -- California
Source: DAI-B 70/07, Dissertation Abstracts International
Subjects: Mathematics
Keywords: Conformal class, Scalar curvature, Yamabe problem
Publication Number: 3364497
ISBN: 978-1-109-24285-0
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