Um breve estudo sobre a massa Gauss-Bonnet-Chern dos gráficos euclidianos - Mota, Alexandre de Sousa (2016)
In this thesis, we investigate the Gauss-Bonnet-Chern (GBC) mass in the class of asymptotically flat Euclidean graphs, of arbitrary codimension, with a closed and planar boundary, possibly empty or disconnected. In the investigation, an integral formula was obtained for this geometric global invariant, expressed in terms of the GBC curvature, the second fundamental and the Newton transformation of the graph, seen as a submnifold of Euclidean space; the contribution of the boundary for the mass is also quantified through an integral formula, expressed in terms of the contact angle of the graph with each hyperplane that contains a connected component of the boundary and a higher order mean curvature of this component, seen as a submanifold of the hyperplane. When this mean curvature is non-negative, the formula can be applied to derive a partial version of the positive GBC mass conjecture, in that graph class. In turn, when the boundary is star-shaped, satisfies an appropriate convexity hypothesis and the contact angle is straight, the formula can be combined with the Alexandrov-Frenchel inequality to derive the partial validity of the Penrose inequality for this notion of mass.