The Area-Angular Momentum-Charge Inequality for Black Holes With Positive Cosmological Constant

TL;DR

This paper proves a conjectured inequality relating the area, angular momentum, and charge of black hole horizons with a positive cosmological constant, showing it is exactly saturated by extreme Kerr-Newman-de Sitter black holes.

Contribution

It establishes the area-angular momentum-charge inequality for stable horizons with a positive cosmological constant, simplifying the proof via convexity of an associated harmonic map energy.

Findings

The inequality is proven for stable apparent horizons with positive cosmological constant.

The inequality is saturated precisely by extreme Kerr-Newman-de Sitter horizons.

The proof utilizes convexity of an area functional related to harmonic maps.

Abstract

We establish the conjectured area-angular momentum-charge inequality for stable apparent horizons in the presence of a positive cosmological constant, and show that it is saturated precisely for extreme Kerr-Newman-de Sitter horizons. As with previous inequalities of this type, the proof is reduced to minimizing an `area functional' related to a harmonic map energy; in this case maps are from the 2-sphere to the complex hyperbolic plane. The proof here is simplified compared to previous results for less embellished inequalities, due to the observation that the functional is convex along geodesic deformations in the target.