Generalized inverse mean curvature flows in spacetime

TL;DR

This paper investigates conditions under which certain surface flows in spacetime preserve the monotonicity of Hawking mass, extending previous work to a broader class of flows relevant to the Penrose inequality.

Contribution

It introduces a generalized framework for inverse mean curvature flows in spacetime, including null and spacelike directions, and extends the weak formulation to this broader class.

Findings

Monotonicity of Hawking mass under generalized flows

Existence results for null flows and partial results for spacelike flows

Extension of Huisken and Ilmanen's weak formulation to spacetime flows

Abstract

Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered for null as well as for spacelike directions. In the null case, local existence of the flow is guaranteed. In the spacelike case, the uniformly expanding condition leaves a 1-parameter freedom, but for the whole family, the embedding functions satisfy a forward-backward parabolic system for which local existence does not hold in general. Nevertheless, we have obtained a generalization of the weak (distributional) formulation of this class of flows, generalizing the corresponding step of Huisken and Ilmanen's proof of the Riemannian Penrose inequality.