The Scalar Curvature Deformation Equation on Locally Conformally Flat Manifolds
نویسنده
چکیده
Abstract. We study the equation ∆gu− n−2 4(n−1)R(g)u+Ku p = 0 (1+ ζ ≤ p ≤ n+2 n−2 ) on locally conformally flat compact manifolds (M, g). We prove the following: (i) When the scalar curvature R(g) > 0 and the dimension n ≥ 4, under suitable conditions on K, all positive solutions u have uniform upper and lower bounds; (ii) When the scalar curvature R(g) ≡ 0 and n ≥ 5, under suitable conditions on K, all positive solutions u with bounded energy have uniform upper and lower bounds. We also give an example to show that the energy bound condition for the uniform estimates in [18] is necessary.
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