Variedades quasi-Einstein compactas com bordo - Sousa, Tiago Gadelha de (2021)
The objective of this work is to study compact quasi-Einstein manifolds with edge. In the first part, we will provide edge estimates and geometric obstruction results. We establish a sharp upper estimate for the edge area of ​​a compact quasi-Einstein manifold with a connected edge assuming a lower bound of the Ricci curvature of the edge. With equality being valid if, and only if, the boundary of the manifold is isometric to a sphere. The result is still valid in the three-dimensional case without the condition of limiting the Ricci curvature of the border. Considering a compact quasi-Einstein manifold with (possibly disconnected) edge and constant scalar curvature, we were also able to obtain a characterization result in terms of surface gravity of the edge components. For the case where the edge is connected, a sharp geometric inequality ensues from this result involving the edge area and the volume of such manifolds, which can also be seen as a result of obstruction. Furthermore, equality occurs if, and only if, the manifold is isometric, unless scaling, to the hemisphere. We conclude the first part of this work by presenting an upper edge estimate for compact quasi-Einstein manifolds with a (possibly disconnected) edge in terms of the Brown-York mass. In the second part of this work, we provide a Böchner-type formula for quasi-Einstein manifolds with a dimension greater than or equal to 3 which allows us to obtain stiffness results assuming a pinched condition involving the traceless Ricci tensor. Furthermore, considering the Yamabe invariant (or the Yamabe constant), which is an important tool in prescribed metric problems, we obtain an integral curvature estimate in terms of the Yamabe constant for 4-dimensional compact quasi-Einstein manifolds with boundary and constant scalar curvature.