The Scalar Curvature Deformation Equation on Locally Conformally Flat Manifolds

Details The Scalar Curvature Deformation Equation on Locally Conformally Flat Manifolds The Scalar Curvature Deformation Equation on Locally Conformally Flat Manifolds

2007-03-20

arxiv.org/abs/math/0703563v1

Mathematics

Differential Geometry

Scientific paper

We study the equation $\Delta_g u -\frac{n-2}{4(n-1)}R(g)u+Ku^p=0 (1+\zeta \leq p \leq \frac{n+2}{n-2})$ on locally conformally flat compact manifolds $(M^n,g)$. We prove the following: (i) When the scalar curvature $R(g)>0$ and the dimension $n \geq 4$, under suitable conditions on $K$, all positive solutions $u$ have uniform upper and lower bounds; (ii) When the scalar curvature $R(g)\equiv 0$ and $n \geq 5$, 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 math.DG/0602636 is necessary.

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