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).