Axisymmetric Jang Equation Analysis

• Axisymmetric Jang equation is a quasilinear PDE that simplifies black hole data analysis by reducing the 3D problem to a 2D orbit-space using symmetry and twist perturbations.
• The formulation enables rigorous proofs of geometric inequalities, notably the Angular Momentum Penrose Inequality, by preserving ADM mass and characterizing boundary blow-up behavior.
• Integration into graph manifolds with controlled asymptotics ensures solution uniqueness and regularity, providing critical insights into black hole stability and gravitational field dynamics.

The axisymmetric Jang equation is a quasilinear partial differential equation central to the geometric analysis of axisymmetric black hole initial data in general relativity. In the setting of vacuum initial data satisfying the dominant energy condition (DEC), it plays a pivotal role in deriving sharp geometric inequalities relating ADM mass, horizon area, and Komar angular momentum. The equation is distinguished by the way axisymmetric structure and twist perturbations enter its analytic form, enabling dimensional reduction and exposing lower-order angular momentum effects as perturbative terms. Its unique boundary and asymptotic properties, as well as its embedding into graph manifolds, facilitate accurate characterization of black hole stability and mass–angular momentum effects, notably culminating in rigorous proofs of the Angular Momentum Penrose Inequality.

1. Axisymmetric Initial Data and Dimensional Reduction

Consider an axisymmetric vacuum initial data set $(M^3, g, K)$ endowed with a Killing field $\eta = \partial_\phi$. The geometry is presented in Weyl–Papapetrou coordinates $(r, z, \phi)$, reducing the problem via symmetry to the 2D orbit-space $\mathcal Q = M/S^1$. The metric is expressed as $$g = e^{2U(r,z)} (dr^2 + dz^2) + \rho(r,z)^2 d\phi^2$$, while the extrinsic curvature decomposes as $K = K^{\rm(sym)} + K^{\rm(twist)}$, with the twist piece defined by $K^{\rm(twist)}_{i\phi} = \frac{1}{2}\rho^2\omega_i$. Here, $\omega = \omega_r\,dr + \omega_z\,dz$ is the twist 1-form. The Jang graph $\Gamma(f) = \{(x, f(x)) : x \in M \}$ in the product $M \times \mathbb R$ naturally inherits axisymmetry if $f$ is $\phi$-independent, reducing the analysis to $(r, z)$ and yielding an effective two-dimensional PDE context (Xu, 7 Dec 2025).

2. The Quasilinear Axisymmetric Jang Equation

The axisymmetric Jang equation arises from equating the mean curvature of the embedded Jang graph with the trace of the extrinsic curvature, $H_{\Gamma(f)} = \mathrm{tr}_{\Gamma(f)}K$. In orbit-space coordinates, it reduces to: $$\bar g^{ij}\left(\frac{\bar\nabla_{ij}\bar f}{\sqrt{1 + |\bar\nabla\bar f|^2}} - \bar K_{ij}\right) - \frac{\bar f^i \bar f^j}{1 + |\bar\nabla\bar f|^2}\left(\frac{\bar\nabla_{ij}\bar f}{\sqrt{1 + |\bar\nabla\bar f|^2}} - \bar K_{ij}\right) - \mathcal T[\bar f] = 0,$$ where $$|\bar\nabla\bar f|^2 = \bar g^{ij}\bar f_{,i}\bar f_{,j}$$, and $\bar f(r, z)$ is independent of $\phi$. The twist perturbation term is given by: $$\mathcal T[\bar f] = \frac{\rho^2}{\sqrt{1 + |\bar\nabla\bar f|^2}} \left(\omega_i\,\bar\nu^i - \frac{\bar f_{,i}\,\omega_j\,\bar f^{,j}}{\sqrt{1 + |\bar\nabla\bar f|^2}}\,\bar\nu^i\right),$$ with the unit-normal projection $$\bar\nu^i = \bar g^{ij}\bar f_{,j}/\sqrt{1 + |\bar\nabla\bar f|^2}$$. The twist term $\mathcal T[\bar f]$ represents a lower-order perturbation, vanishing asymptotically both near the blow-up surface and at spatial infinity (Xu, 7 Dec 2025).

3. Geometric Ansatz and Graph Embedding

The Jang construction reformulates the geometric Cauchy problem by embedding the axisymmetric initial data into the graph $\Gamma(f)$ within $M \times \mathbb R$. The induced metric on this graph is $\tilde g = g + df \otimes df$, preserving axisymmetry if $f$ is $\phi$-independent. This leads to a reduction of the analytic and geometric complexity to the $(r, z)$ coordinate plane, allowing direct application of tools from two-dimensional Riemannian geometry and facilitating the blow-up analysis near boundaries (Xu, 7 Dec 2025).

4. Twist Potential and Perturbative Structure

The twist 1-form $\omega$ on $\mathcal Q$ encapsulates the angular momentum content via $K^{\rm(twist)}_{i\phi} = \frac{1}{2}\rho^2\omega_i$, with $d\omega = 0$ enforced by the vacuum momentum constraint. While locally a twist potential $\psi$ may be introduced such that $d\psi = \star_{\mathcal Q}\omega$, the axisymmetric Jang equation's analytic structure is only sensitive to $\omega$. The term $\mathcal T[\bar f]$ is uniformly bounded on compact sets, decays as $O(r^{-2})$ at spatial infinity, and vanishes linearly with the distance to $\Sigma$, confirming its status as a lower-order perturbation in both the asymptotic and singular limits. This feature is instrumental in ensuring solution existence, regularity, and mass monotonicity (Xu, 7 Dec 2025).

5. Boundary Behavior and Asymptotics

The blow-up surface $\Sigma \subset M$, corresponding to the outermost stable marginally outer trapped surface (MOTS), induces a logarithmic divergence in the Jang solution: $$f(s, y) = C_0\,\ln s^{-1} + A(y) + O(s^\alpha),\quad C_0 = \frac{1}{2}|\theta^-| > 0,$$ where $\theta^- < 0$ is the inward null-expansion on $\Sigma$. The induced metric assumes a cylindrical end structure in the limit, with $\tilde g = dt^2 + g_\Sigma + O(e^{-\beta t})$, $t = -\ln s$, $\beta > 0$. At spatial infinity, the solution satisfies $f(r, z) = O(r^{1 - \tau + \epsilon})$, with $\tau > 1/2$ and controlled decay for capillarity regularization, ensuring preservation of the ADM mass. Such asymptotics are essential for preserving global mass inequalities and enabling the passage to the limit in variational methods (Xu, 7 Dec 2025).

6. Analytic Framework: Existence and Uniqueness

Solution existence and uniqueness for the axisymmetric Jang equation rely on five main analytic steps:

1. Equivariant Reduction: Converts the full 3D Jang equation to the 2D orbit-space, incorporating the twist term $\mathcal T[\bar f]$.
2. Lower-Order Verification: Confirms boundedness and decay properties for $\omega$ and the twist term.
3. Barrier Construction: Utilizes supersolutions and subsupersolutions for spatial infinity and near $\Sigma$, respectively, applying the Schoen–Yau stability barrier method for enforced blow-up.
4. Perron/Capillarity Method: Regularizes the equation, solves the capillary boundary problem, applies Schauder estimates, and passes limits for singularity formation.
5. Asymptotics and Uniqueness: Han–Khuri analysis provides sharp asymptotics, while the maximum principle ensures uniqueness up to vertical translations of $f$. Crucially, mass is preserved with $M_{ADM}(g+df^2) \le M_{ADM}(g)$ and equality only in time-symmetric settings (Xu, 7 Dec 2025).

7. Role in the Angular Momentum Penrose Inequality

The axisymmetric Jang equation is instrumental in proving the Angular Momentum Penrose Inequality for black holes. Specifically:

• The solution defines a new Riemannian manifold $(M, \tilde g)$ with non-negative distributional scalar curvature under DEC via the Bray–Khuri identity, a cylindrical end at $\Sigma$, and ADM mass no greater than the initial mass.
• Subsequent solution of a modified Lichnerowicz equation yields a conformal metric with nonnegative scalar curvature and pointwise conformal factor bound.
• The Agostiniani–Mazzieri–Oronzio flow is run on this background, with the cylindrical end structure ensuring the Fredholm property for $p \to 1$.
• Exact conservation of Komar angular momentum $J$ along flow surfaces follows from closedness of $\alpha_J = \frac{1}{8\pi}K(\eta,\cdot)^\flat$ and application of Stokes’ theorem, due to $d\alpha_J = 0$.
• The monotonicity of the combined area–angular-momentum functional

$$\mathcal M_{1,J}(t) = \sqrt{\frac{A(t)}{16\pi} + \frac{4\pi J^2}{A(t)}}$$

is ensured by Bray–Khuri divergence identities.

• Enforcement of the Dain–Reiris sub-extremal inequality $A(t) \ge 8\pi|J|$ guarantees non-decreasing behavior of the functional to the ADM mass limit, establishing

$$M_{ADM} \ge \sqrt{\frac{A}{16\pi} + \frac{4\pi J^2}{A}},$$

with equality precisely for Kerr initial data (Xu, 7 Dec 2025).

A plausible implication is that the axisymmetric Jang equation, by encoding twist as a lower-order perturbation and facilitating the geometric reduction, provides a universal analytic engine for sharp black hole inequalities in axisymmetric contexts, with significant ramifications for cosmic censorship, black hole thermodynamics, and gravitational wave astrophysics.