Abstract
The local black holes describe physical situations involving a black hole surrounded by a finite vacuum region and then by matter and fields. The stationary and axisymmetric local black holes belong into two classes, the spherical and the toroidal ones, depending on the topology of their horizon. For the static black holes their metric tensors are given explicitly in terms of Legendre polynomials. For the stationary local black holes the problem is formulated interms of the Ernst potential of the rotational Killing field and the appropriate asymptotic conditions on the horizon are determined.
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