Evolution of Hawking mass under hypersurface-restricted expanding flows

TL;DR

This paper numerically investigates how the Hawking mass evolves under expanding flows for perturbed surfaces in Minkowski spacetime, confirming monotonicity robustness and identifying numerical challenges.

Contribution

It provides the first numerical analysis of Hawking mass evolution for nonspherical surfaces under expanding flows, demonstrating monotonicity persistence and highlighting computational difficulties.

Findings

Monotonicity of Hawking mass persists for certain nonspherical perturbations.

Robustness of monotonicity under variations in perturbation amplitude and frequency.

Identification of numerical instabilities in the flow simulations.

Abstract

We present a numerical study of the evolution of the Hawking mass for closed nonspherical surfaces evolved under a class of expanding flows in Minkowski spacetime. Although formal monotonicity of the Hawking mass under smooth inverse mean curvature flow is well established in the Riemannian setting, comparatively little is known about the robustness of this behavior in discrete numerical implementations applied to explicitly embedded surfaces away from exact symmetry. We consider surfaces defined by small spherical harmonic perturbations of a round sphere and evolve them under an in-slice, time-flat flow analogous to inverse mean curvature flow. We examine the behaviour of the Hawking mass under the flow and find that monotonicity persists for a class of nonspherical perturbations and is robust under variations in perturbation amplitude and angular frequency. We also identify regimes in which numerical instabilities arise, highlighting practical challenges associated with extending such flows beyond simple symmetry assumptions. These results provide a concrete computational testbed for future investigations of uniformly expanding flows and quasi-local mass in more general spacetime settings.