We determine all infinitesimal transverse deformations of extreme horizons in Einstein-Maxwell theory that preserve axisymmetry. In particular, we show that the general static transverse deformation of the AdS(2) X S(2) near-horizon geometry is a two-parameter family, which contains the known extreme charged, accelerating, static black hole solution held in equilibrium by an external electric or magnetic field (Ernst solution) and a special case of the extreme Kerr-Newman-Melvin solution. More generally, we find a three-parameter family of deformations of the extreme Kerr-Newman horizon, which contains the extreme Kerr-Newman-Melvin solution and a rotating generalisation of the Ernst solution. We also consider vacuum gravity with a cosmological constant and prove uniqueness of axisymmetric transverse deformations of the extreme Kerr-AdS horizon. Finally, we completely classify transverse deformations of extreme horizons in three-dimensional Einstein-Maxwell theory with a negative cosmological constant.

中文翻译：

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极端视界附近的电真空时空

我们在保持轴对称的爱因斯坦-麦克斯韦理论中确定了极端视界的所有无穷小横向变形。特别是，我们表明 AdS(2) XS(2) 近视距几何的一般静态横向变形是一个双参数族，其中包含已知的极端带电、加速、静态黑洞解，通过外部电场或磁场（Ernst 解）和极端 Kerr-Newman-Melvin 解的特例。更一般地说，我们找到了极端 Kerr-Newman 视界的三参数变形族，其中包含极端 Kerr-Newman-Melvin 解和 Ernst 解的旋转推广。我们还考虑了具有宇宙学常数的真空重力，并证明了极端 Kerr-AdS 视界的轴对称横向变形的唯一性。