A new approach towards the construction of initial data in general relativity with positive Yamabe invariant and arbitrary mean curvature

TL;DR

This paper introduces a novel fixed point approach to constructing initial data in general relativity with positive Yamabe invariant, ensuring solution uniqueness and explicit construction under certain volume bounds.

Contribution

It replaces the Schauder fixed point theorem with the Banach fixed point theorem in the conformal method, providing guarantees of uniqueness and an explicit solution construction.

Findings

Guarantees uniqueness of solutions under volume bounds

Provides an explicit construction of initial data

Simplifies the proof of existence using Banach fixed point theorem

Abstract

This paper revisits the classical construction of initial data using the conformal method, as originally proposed by Holst, Nagy, and Tsogtgerel and later refined by Maxwell. We demonstrate that the existence of the solution can be proven using the Banach fixed point theorem, whereas the original proof relied on the Schauder fixed point theorem. This new approach has two main advantages: it guarantees the uniqueness of the solution to the equations of the conformal method as soon as one imposes a bound on the physical volume of it and it provides an explicit construction of the solution.