# AdS$_3$ to dS$_3$ transition in the near horizon of asymptotically de Sitter solutions

## TL;DR

This paper explores how the near horizon geometries of EVH black holes in Einstein-$
Lambda$ theory can transition between AdS$_3$, flat, and dS$_3$ spacetimes, revealing new insights into their thermodynamics and potential holographic duals.

## Contribution

It demonstrates the interpolation of near horizon geometries between AdS$_3$, flat, and dS$_3$ in EVH black holes, and analyzes their thermodynamics and holographic implications.

## Key findings

- Near horizon EVH Kerr geometries interpolate between AdS$_3$, flat, and dS$_3$.
- The near horizon of EVH cosmological solitons is always dS$_3$ and regular.
- Thermodynamics of the 3d geometries matches that of the original EVH black holes.

## Abstract

We consider two solutions of Einstein-$\Lambda$ theory which admit the extremal vanishing horizon (EVH) limit, odd-dimensional multi-spinning Kerr black hole (in the presence of cosmological constant) and cosmological soliton. We show that the near horizon EVH geometry of Kerr has a 3d maximally symmetric subspace whose curvature depends on rotational parameters and the cosmological constant. In the Kerr-dS case, this subspace interpolates between AdS$_3$, 3d flat and dS$_3$ by varying rotational parameters, while, the near horizon of the EVH cosmological soliton always has a dS$_3$. The feature of the EVH cosmological soliton is that it is regular everywhere on the horizon. In the near EVH case, these 3d parts turn into the corresponding locally maximally symmetric spacetimes with a horizon: Kerr-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 2d CFT dual to the near horizon of EVH solutions.