Rigidity Theorems of conformal class on compact manifolds with boundary
  Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of metrics in a same conformal class that fixes the boundary and satisfy certain integral conditions on the the scalar curvatures and the mean curvatures on the boundary. No condition on the first eigenvalues of operators is need. 
Ezequiel Barbosa, Heudson Mirandola, Feliciano Vitorio
math.DG