Volume comparison and the sigma_k-Yamabe problem

Details Volume comparison and the sigma_k-Yamabe problem Volume comparison and the sigma_k-Yamabe problem

2002-10-18

arxiv.org/abs/math/0210302v2

Final version: Advances in Mathematics 187 (2004), no. 2, 447-487

Mathematics

Differential Geometry

37 pages; v2 title change, to appear in Advances in Math

Scientific paper

In this paper we study the problem of finding a conformal metric with the property that the k-th 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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