Some characterizations of compact Einstein-type manifolds

In this work, we investigate the geometry and topology of compact Einstein-type manifolds with nonempty boundary. First, we prove a sharp boundary estimate and as consequence we obtain, under certain hypotheses, that the Hawking mass is bounded from below in terms of area. Then we give a topological classification for its boundary. Finally, we deduce some classification results for compact Einstein-type manifolds with positive constant scalar curvature and assuming a pointwise inequality for the traceless Ricci tensor.