Volume comparison and the sigma_k-Yamabe problem
Volume comparison and the sigma_k-Yamabe problem
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.
37 pages; v2 title change, to appear in Advances in Math
- University of Notre Dame United States
- Massachusetts Institute of Technology United States
Mathematics - Differential Geometry, prescribing curvature, 53A30, Yamabe, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Nonlinear elliptic equations, conformal geometry, 35J60, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), FOS: Mathematics, 35J60; 53A30, Analysis of PDEs (math.AP)
Mathematics - Differential Geometry, prescribing curvature, 53A30, Yamabe, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Nonlinear elliptic equations, conformal geometry, 35J60, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), FOS: Mathematics, 35J60; 53A30, Analysis of PDEs (math.AP)
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