A-PRIORI ESTIMATES FOR PRESCRIBING SCALAR CURVATURE EQUATIONS

We obtain a priori estimates for solutions to the prescribing scalar c urvature equation [GRAPHICS] on S-n for n greater than or equal to 3. There have been a series of results in this respect. To obtain a prior i estimates people required that the function R(x) be positive and bou nded away from 0. This technical assumption has been used by many auth ors for quite a few years. It is due to the fact that the standard blo wing-up analysis fails near R(x) = 0. The main objective of this paper is to remove this well-known assumption. Using the method of moving p lanes, we are able to control the growth of the solutions in the regio n where R is negative and in the region where R is small, and thus obt ain a priori estimates on the solutions of (1) for a general function R which is allowed to change signs.