On The Conformal Schouten Functional And Its Critical Metrics

Let M~n be a given compact, smooth n-manifold without boundary, and T be a con-formal class on M~n. We consider the normalized conformal Schouten functional on the conformal class T, defined bywhereis Schouten tensor.By calculating the first variation of S_T, we proveTheorem A. Let T be a conformal class on a 4- dimensional compact manifold M~4. Then a metric g∈T is a critical point of the conformal Schouten functional S_T if and only if the scalar curvature R of g is constant.Furthermore, by calculating the second variation of S_T, We can proveTheorem B. Every critical metrics of nonpositive constant scalar curvature are stable for S_T on a 4-dimensional compact manifold.