Hawking Mass Monotonicity for Initial Data Sets

TL;DR

This paper generalizes Hawking mass monotonicity to initial data sets satisfying the dominant energy condition by introducing PDE systems modeling double-null foliations, with implications for geometric analysis.

Contribution

It introduces new PDE systems for initial data sets that enable the extension of Hawking mass monotonicity results to broader contexts.

Findings

Established existence of the PDE systems for initial data sets.

Proved monotonicity of Hawking mass under the new framework.

Derived geometric applications from the generalized monotonicity.

Abstract

We introduce new systems of PDE on initial data sets $(M,g,k)$ whose solutions model double-null foliations. This allows us to generalize Geroch's monotonicity formula for the Hawking mass under inverse mean curvature flow to initial data sets satisfying the dominant energy condition. We study the existence theory of these systems and give geometric applications.