Volume comparison and the σk-Yamabe problem |
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Authors: | Matthew J Gursky Jeff A Viaclovsky |
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Affiliation: | a Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA b Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA |
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Abstract: | In this paper we study the problem of finding a conformal metric with the property that the kth elementary symmetric polynomial of the eigenvalues of its Weyl-Schouten tensor is constant. A new conformal invariant involving maximal volumes is defined, and this invariant is then used in several cases to prove existence of a solution, and compactness of the space of solutions (provided the conformal class admits an admissible metric). In particular, the problem is completely solved in dimension four, and in dimension three if the manifold is not simply connected. |
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Keywords: | 53C21 35J60 |
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