Mean curvature flow techniques provide a new approach to the 50 years-old Cosmic No-Hair conjecture in General Relativity. Some of the recent achievements are conditional on the existence of immortal solutions for the mean curvature flow PDE, with nice regularity properties.
    In our work, we describe a method for constructing such solutions in the de Sitter space, a setting of particular interest.
    In the talk, we'll define the objects and the setting, provide a heuristic picture of the program, and survey the construction we've contributed.