---
author:
- 'Hideo [Kodama]{}[^1]'
title: |
     Perturbative Uniqueness of Black Holes near the Static Limit\
    in All Dimensions 
---

Introduction
============

In recent years, inspired by proposals of various higher-dimensional universe models, the investigation of black holes in higher dimensions has become quite active. One of the main issues in these investigations is whether the celebrated black hole uniqueness theorem in four dimensions[@Heusler.M1996B; @Heusler.M1998; @Chrusciel.P1996] also holds in higher dimensions.

Results obtained to this time concerning this issue are quite intriguing[@Kodama.H2004A]. On one hand, it has been shown that Israel’s theorem on the rigidity and uniqueness of static black holes also holds in higher dimensions, that is, the asymptotically flat and static regular black hole solutions are always spherically symmetric for the vacuum system. This rigidity theorem has been extended to the electro-vacuum and Einstein-Maxwell-Dilaton systems with non-degenerate horizons.

On the other hand, concerning rotating black holes, we now have two families of asymptotically flat regular vacuum solutions with different horizon topologies in five dimensions, the Myers-Perry solution and the Emparan-Reall black ring solution. The former is a higher-dimensional analogue of the Kerr solution, and its black hole surface is diffeomorphic to a sphere, while the latter has no 