In this talk I will reconsider the no-hair theorems trying to understand two main aspects. First of all, no-hair theorems are always formulated in vacuum. However, it is legitimate to ask about the role that external matter can play in the structure of black hole solutions, namely whether for a given external matter distribution there exists more than one family of black holes that are compatible with such matter distribution. Second, it is interesting to consider what happens if one substitutes the assumption of a smooth event horizon (zero redshift surface) by the existence of an arbitrary but small finite redshift surface. This sheds light on the origin of the theorems and allows one to consider situations in which we have an almost perfect mimicker of a black hole.To address these questions, I will discuss the structure of static vacuum Einstein equations, putting a special emphasis on the elliptic problem that the redshift function obeys. Then, I will consider a simplified scenario, in which I restrict myself to consider axisymmetric spacetimes since in that case the elliptic problem reduces to a Laplacian problem in Euclidean 3D space. To answer the first question, I will first discuss how external matter content can deform the geometry of the horizon and discuss how the contribution of the black hole and the and the external matter to the multipolar structure can be in a precise sense decoupled. To answer the second question, I will assume the existence of a surface of arbitrary small redshift and see what are the constraints that we can obtain on the multipolar structure of the object if we impose that the curvature remains bounded as this minimum redshift tends to zero (horizon limit).

https://indico.math.cnrs.fr/event/10967/