About an integral inequality and rigidity of m -quasi-Einstein manifolds

In this short note, we apply a Reilly-type identity to obtain an integral inequality for the boundary of a bounded domain in a m- quasi-Einstein manifold. Furthermore, a rigidity characterization for the whole manifold is obtained in the equality case. Such a result extends a previous one obtained in the context of static manifolds and it is related to the stability of Wang–Yau Energy.