The concept of mass for time-symmetric initial data has been extensively explored and is now a cornerstone in the study of contemporary Mathematical General Relativity, especially in relation to spacetimes with zero or negative cosmological constants. However, the case of a positive cosmological constant presents a distinct challenge: the renowned counterexamples to the Min-Oo conjecture by Brendle, Marques, and Neves highlight that even the rigidity statement in a potential positive mass theorem has not been correctly identified yet in this context. In this presentation, I will propose a novel approach to overcome this issue, leading to insights on a new notion of mass and to a characterization of the de-Sitter spacetime. This is a joint work with Virginia Agostiniani and Lorenzo Mazzieri.
