On some conformally invariant fully nonlinear equations, II. Liouville, Harnack and Yamabe
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Let (M, g) be an n-dimensional compact smooth Riemannian manifold (without boundary). For n=2, we know from the uniformization theorem of Poinca% that there exist metrics that are pointwise conformal to g and have constant Gauss curvature. For n~>3, the well-known Yamabe conjecture states that there exist metrics which are pointwise conformal to g and have constant scalar curvature. The Yamabe conjecture is proved through the work of Yamabe [65], Trudinger [58], Aubin [2] and Schoen [53]. The Yamabe and related problems have attracted much attention in the last 30 years or so, see, e.g., [57], [3] and the references therein. Important methods and techniques in overcoming loss of compactness have been developed in such studies, which also play important roles in the research of other areas of mathematics. For n~>3, let ~=u4/(n-2)g, where u is some positive function on M. The scalar curvature RO of ~ can be calculated as
Subfields: Applied Mathematics, Computational Theory and Mathematics
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Citations in the corpus
- Local gradient estimates of solutions to some conformally invariant fully nonlinear equations (2006, methodology): “4 is along the line of [26, 27], which makes use of ideas developed in [19, 23] in treating the isolated singularity of u at 1.”
- Variational status of a class of fully nonlinear curvature prescription problems (2006, background): “Recently there has been significant interest and progress in the study of the so-called σkYamabe problem and related curvature prescription problems, see for example [16,7,12,14, 11].”
- Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary (2006, background): “6) and f |∂ = 0 has been proved in [27,29] on compact locally conformally flat manifolds without boundary.”
- Recent developments in elliptic partial differential equations of Monge-Ampere type (2006, background): “1 when k = 2, n = 4 was proved in the pioneering work of Chang, Yang and Gursky [6], while the locally conformally flat case was proved by Guan and Wang [15] and Li and Li [19], [20].”
- Remark on some conformally invariant integral equations: the method of moving spheres (2003, methodology): “The method of moving spheres has been used in [22]-[25].”