In this talk I want to analyze axially symetric, vacuum, spacetimes. In particular, I want to present new ingredients that I hope will open new possibilities for controlling the evolution of such systems in problems related with the linear and non linear stability of the extreme Kerr black hole.

These new ingredients are related with the total mass in General Relativity. In axisymmetric evolution of isolated systems there exists a gauge such that the total mass can be written as a positive definite integral on the spacelike hypersurfaces of the foliation and the integral is constant along the evolution. Moreover, extreme Kerr is a global minimum of this integral. In particular, cylindrical geometries (also called trumpet initial data) appears naturally in this context. I will also discuss some recent results on this kind of initial data.

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