Variable cosmological term - geometry and physics
  We describe the dynamics of a cosmological term in the spherically symmetric case by an r-dependent second rank symmetric tensor \Lambda_{\mu\nu} invariant under boosts in the radial direction. The cosmological tensor \Lambda_{\mu\nu} represents the extension of the Einstein cosmological term \Lambda g_{\mu\nu} which allows a cosmological constant be variable. This possibility is based on the Petrov classification scheme and Einstein field equations in the spherically symmetric case. The inflationary equation of state is satisfied by the radial pressure p_r^{\Lambda}=-\rho^{\Lambda}. The tangential pressure is calculated from the conservation equation \Lambda^{\mu}_{\nu;\mu}=0. The solutions to the Einstein equations with cosmological term \Lambda_{\mu\nu} describe several types of globally regular self-gravitating vacuum configurations including vacuum nonsingular black holes. In this case the global structure of space-time contains an infinite set of black and white holes whose singularities are replaced with the value of cosmological constant of the scale of symmetry restoration, at the background of asymptotically flat or asymptotically de Sitter universes. We outline \Lambda white hole geometry and estimate probability of quantum birth of baby universes inside a \Lambda black hole. In the course of Hawking evaporation of a \Lambda black hole, a second-order phase transition occurs, and globally regular configuration evolves towards a self-gravitating particlelike structure at the background of the Minkowski or de Sitter space. 
Irina Dymnikova (University of Warmia and Mazury, Poland)