Numerical investigation of the generalized Jang equation coupled to conformal flow of metrics

TL;DR

This work numerically investigates the generalized Jang equation coupled to Bray's conformal flow of metrics in a spherically symmetric, time-symmetric setting to assess whether Jaracz-type finite-radius breakdowns persist. By reducing the system to a tractable set of ODEs with the conformal factor $u(r)=e^{v(r)}$, warping factor $\phi(r)=u(r)$, and Jang slope $Q(r)=\frac{f'(r)}{\sqrt{1+f'(r)^2}}$, the authors perform a rigorous numerical study validated against Schwarzschild data. They find that $|Q|$ approaches the degeneracy barrier $1$ only asymptotically and that $Q'(r)$ remains bounded, indicating no finite-radius breakdown, with this behavior robust to controlled perturbations of the warping factor. These results suggest that coupling to conformal flow alters the obstruction mechanism identified by Jaracz, supporting the viability of the Jang/conformal flow approach for proving the Penrose inequality, though a full proof remains open and warrants further analysis.

Abstract

A recent result of Jaracz has established nonexistence of global solutions to the coupled generalized Jang equation and zero divergence system which satisfy the asymptotic conditions needed to prove the Penrose conjecture by identifying a breakdown mechanism for the Jang slope at finite radius. In this work, we investigate whether a similar obstruction arises when the generalized Jang equation is instead coupled to the conformal flow of metrics. Restricting to spherical symmetry and time-symmetric initial data, we formulate a numerically tractable version of the Jang/conformal flow system. Our numerical results show no evidence of a finite radius breakdown analogous to that observed by Jaracz. Instead, the Jang slope remains regular and approaches its limiting value asymptotically. This behavior persists under controlled perturbations of the warping factor, indicating robustness of the observed phenomenon. These findings suggest that coupling to conformal flow of metrics alters the obstruction mechanism present in the Jang/zero divergence system, and hence that this system may still be viable for proving the Penrose conjecture.

Figures (6)

• Figure 1: Numerical solution of the harmonic equation $\Delta v = 0$ in spherical symmetry compared with the exact Schwarzschild solution $v(r) = -1 + r_h/r$. The numerical solution is obtained using a second-order finite difference discretization with horizon and asymptotic boundary conditions. Excellent agreement is observed across the domain, validating the numerical implementation of the conformal flow equation.
• Figure 2: Near-horizon behavior of the conformal factor $u(r)$ for Schwarzschild initial data at increasing resolution $N$. The numerical solutions satisfy the expected horizon condition $u(r_h) =1$. Although the curves do not coincide pointwise near the horizon due to differing grid spacings, the limiting behavior as $r \rightarrow r_h^+$ is stable under refinement and no spurious boundary layers are observed.
• Figure 3: Numerical solution for the Jang slope $Q(r)$ in the spherically symmetric conformal flow system. The slope increases monotonically from the horizon value $Q(r_h)=0$ and approaches the limiting value $|Q|=1$ only asymptotically. No finite radius breakdown is observed, in contrast to the Jang/zero divergence system.
• Figure 4: Radial derivative $Q'(r)$ corresponding to the solution shown in Fig. 3. The derivative remains bounded and decays to zero as $|Q|\to 1$, indicating saturation of the Jang slope, rather than blow-up at finite radius.
• Figure 5: Jang slope $Q(r)$ for the perturbed warping factors $\phi_\varepsilon$ defined in equation (6), shown for several values of $\varepsilon$. All solutions exhibit saturation at $|Q|=1$ only asymptotically, with no finite radius breakdown.