A new mass for asymptotically flat manifolds

In this paper we introduce a mass for asymptotically flat manifolds by using the Gauss–Bonnet curvature. We first prove that the mass is well-defined and is a geometric invariant, if the Gauss–Bonnet curvature is integrable and the decay order τ   satisfies τ>n−43. Motivated by an elegant idea of Lam [35], we then show a positive mass theorem for this new mass for asymptotically flat graphs over RnRn. Moreover we obtain various Penrose type inequalities in this case. In these Penrose type inequalities we establish relationship between the new mass and various geometric integrals.