in this talk, we establish an analytic foundation for a fully non-linear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds with positive scalar curvature. this equation arises from conformal geometry. as application, we prove that, if a compact 3-dimensional manifold $m$ admits a riemannian metric with positive scalar curvature and $\int \sigma_2\ge 0$, then topologically $m$ is a quotient of sphere.