Some Compactness Results Related to Scalar Curvature Deformation

Details Some Compactness Results Related to Scalar Curvature Deformation Some Compactness Results Related to Scalar Curvature Deformation

2006-02-27

arxiv.org/abs/math/0602636v2

Mathematics

Differential Geometry

Scientific paper

Motivated by the prescribing scalar curvature problem, we study the equation $\Delta_g u +Ku^p=0 (1+\zeta \leq p \leq \frac{n+2}{n-2})$ on locally conformally flat manifolds $(M,g)$ with $R(g)=0$. We prove that when $K$ satisfies certain conditions and the dimension of $M$ is 3 or 4, any solution $u$ of this equation with bounded energy has uniform upper and lower bounds. Similar techniques can also be applied to prove that on 4-dimensional scalar positive manifolds the solutions of $\Delta_gu-\frac{n-2}{4(n-1)}R(g)u+Ku^p=0, K>0, 1+\zeta \leq p \leq \frac{n+2}{n-2}$ can only have simple blow-up points.

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