Intrinsic rigidity of extremal horizons

TL;DR

The paper proves an intrinsic rigidity for compact cross-sections of vacuum extremal horizons by showing every nontrivial solution of the horizon equation admits a Killing vector $K$ with $K^\flat=\Gamma X^\flat+\mathrm{d}\Gamma$, where $\Gamma>0$ and $K$ is constructed via a principal eigenfunction of a related elliptic operator. A central tensor identity enables an integral argument that forces $K$ to be a genuine Killing field, and for $\lambda\le 0$ it also yields $[K,X]=0$, extending to near-horizon symmetries. In 2D, the horizon equation reduces to a fourth-order PDE for the Kähler potential; on $S^2$ this equation is explicitly solvable, and with axial symmetry the solution reduces to a linear ODE, recovering the extremal Kerr horizon (with cosmological constant) as the unique nontrivial cross-section in the gradient-free case. The work also proves a near-horizon symmetry enhancement to $\mathfrak{sl}(2)\times\mathfrak{u}(1)$ in the vacuum with $\lambda\le 0$, and provides a detailed framework linking intrinsic horizon geometry to Kerr-type and AdS$_2$-structured near-horizon geometries. Altogether, these results complete the intrinsic classification of vacuum extremal horizons with 2D cross-sections and illuminate the role of Killing symmetries in higher dimensions and with a cosmological constant.

Abstract

We prove that the intrinsic geometry of compact cross-sections of any vacuum extremal horizon must admit a Killing vector field. If the cross-sections are two-dimensional spheres, this implies that the most general solution is the extremal Kerr horizon and completes the classification of the associated near-horizon geometries. The same results hold with a cosmological constant. Furthermore, we also deduce that any non-trivial vacuum near-horizon geometry, with a non-positive cosmological constant, must have a Lie algebra of Killing vector fields that contains $\mathfrak{sl}(2)\times \mathfrak{u}(1)$ in all dimensions under no symmetry assumptions. We also show that, if the cross-sections are two-dimensional, the horizon Einstein equation is equivalent to a single fourth order PDE for the Kähler potential, and that this equation is explicitly solvable on the sphere if the corresponding metric admits a Killing vector.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4