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

TL;DR

This study numerically examines the generalized Jang equation coupled with conformal flow of metrics, finding no finite radius breakdown and suggesting this approach may be promising for proving the Penrose conjecture.

Contribution

The paper introduces a numerical investigation of the coupled Jang/conformal flow system, revealing stable behavior and potential advantages over previous models for the Penrose conjecture.

Findings

No finite radius breakdown observed in the coupled system.

Jang slope remains regular and approaches a limit asymptotically.

Robustness of behavior under perturbations of the warping factor.

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.