We consider two solutions of Einstein-$\mathrm{\Lambda }$ theory which admit the extremal vanishing horizon (EVH) limit, odd-dimensional multispinning Kerr black hole (in the presence of cosmological constant) and cosmological soliton. We show that the near horizon EVH geometry of Kerr has a three-dimensional maximally symmetric subspace whose curvature depends on rotational parameters and the cosmological constant. In the Kerr-dS case, this subspace interpolates between ${\mathrm{AdS}}_3$, three-dimensional flat and ${\mathrm{dS}}_3$ by varying rotational parameters, while the near horizon of the EVH cosmological soliton always has a ${\mathrm{dS}}_3$. The feature of the EVH cosmological soliton is that it is regular everywhere on the horizon. In the near EVH case, these three-dimensional parts turn into the corresponding locally maximally symmetric spacetimes with a horizon: Kerr-${\mathrm{dS}}_3$, flat space cosmology or BTZ black hole. We show that their thermodynamics match with the thermodynamics of the original near EVH black holes. We also briefly discuss the holographic two-dimensional CFT dual to the near horizon of EVH solutions.

• Received 15 June 2017

DOI:https://doi.org/10.1103/PhysRevD.96.044004