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The scalar-curvature problem on the standard three-dimensional sphere. (English) Zbl 0722.53032
The authors study the problem of finding a metric conformally equivalent to the standard metric on the sphere $$S^ 3$$ and with prescribed scalar curvature K. The difficulty consists in the failure of the Palais-Smale condition of the corresponding variational problem. This difficulty is overcome in the paper under certain (non-degeneracy) assumptions on K.
Reviewer: W.Ballmann (Bonn)

MSC:
 53C20 Global Riemannian geometry, including pinching
Full Text:
References:
 [1] Kazdan, J; Warner, F, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature, Ann. of math., 101, 317-331, (1975) · Zbl 0297.53020 [2] \scJ. P. Bourguignon and J. P. Ezin, Scalar curvature functions in a conformal class of metrics and conformal transformations, in preparation. · Zbl 0622.53023 [3] Escobar, J; Schoën, R, Conformal metrics with prescribed scalar curvature, Invent. math., 86, 243-254, (1986) · Zbl 0628.53041 [4] Chang, S.Y; Yang, P, Conformal deformations of metrics on S2, J. differential geom., 27, 259-296, (1988) [5] \scS. Y. Chang and P. Yang, Prescribing Gaussian curvature on S2, Acta Math., in press. [6] Chen, W.X; Ding, W, Scalar curvature on S2, Trans. amer. math. soc., 303, 365-382, (1987) [7] Hong, C, A best constant and the Gaussian curvature, (), 737-747 · Zbl 0603.58056 [8] Bahri, A, Pseudo-orbits of contact forms, () · Zbl 0677.58002 [9] Bahri, A; Coron, J.M, Vers une théorie des points critiques à l’infini, () · Zbl 0585.58004 [10] \scA. Bahri, Critical points at infinity in some variational problems, in “Research Notes in Mathematics,” Longman-Pitman, London, in press. · Zbl 0676.58021 [11] Bahri, A; Coron, J.M, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. pure appl. math., 41, 253-294, (1988) · Zbl 0649.35033 [12] Brezis, H; Coron, J.M, Convergence of solutions of H-systems on how to blow bubbles, Arch. rational mech. anal., 89, 21-56, (1985) · Zbl 0584.49024 [13] Sacks, J; Uhlenbeck, K, The existence of minimal immersions of 2-spheres, Ann. of math., 113, 1-24, (1981) · Zbl 0462.58014 [14] Wente, H, Large solutions to the volume constrained platea problem, Arch. rational mech. anal., 75, 59-77, (1980) · Zbl 0473.49029 [15] Struwe, M, A global compactness result for elliptic boundary value problems involving nonlinearities, Math. Z., 187, 511-517, (1984) · Zbl 0535.35025 [16] Lions, P.L; Lions, P.L, The concentration compactness principle in the calculus of variations, Rev. mat. iberoamericana, Rev. mat. iberoamericana, 1, 45-121, (1985) · Zbl 0704.49006 [17] Taubes, C, Path connected Yang-Mills moduli spaces, J. differential geom., 19, 337-392, (1984) · Zbl 0551.53040 [18] Moser, J, On a nonlinear problem in differential geometry, () [19] Sedlacek, S, A direct method for minimizing the Yang-Mills functional over 4-manifolds, Comm. math. phys., 86, 515-527, (1982) · Zbl 0506.53016 [20] \scA. Bahri and H. Brezis, Nonlinear elliptic equations on Riemannian manifolds with the Sobolev critical exponent, to appear. · Zbl 0863.35037 [21] Siu, Y.T; Yau, S.T, Compact Kähler manifolds of positive bisectional curvature, Invent. math., 59, 189-204, (1980) · Zbl 0442.53056
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