On the σ2-scalar curvature

Abstract. In this paper, we establish an analytic foundation for a fully non-linear equation σ2 σ1 = f on manifolds with positive scalar curvature and apply it to give a (rough) classification of such manifolds. A crucial point is a simple observation that this equation is a degenerate elliptic equation without any condition on the sign of f and it is elliptic not only for f> 0 but also for f < 0. By defining a Yamabe constant Y2,1 with respect to this equation, we show that a positive scalar curvature manifold admits a conformal metric with positive scalar curvature and positive σ2-scalar curvature if and only if Y2,1> 0. We give a complete solution for the corresponding Yamabe problem. Namely, let g0 be a positive scalar curvature metric, then in its conformal class there is a conformal metric with σ2(g) = κσ1(g), for some constant κ. Using these analytic results, we give a rough classification of the space of manifolds with positive scalar curvature metrics. 1