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

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.

💡 Research Summary

The paper revisits the conformal method for constructing initial data sets in general relativity, originally introduced by Holst, Nagy, and Tsogtgerel and later refined by Maxwell. While the classical approach proved existence of solutions using the Schauder fixed‑point theorem, it did not guarantee uniqueness and was non‑constructive. The authors replace this with the Banach contraction mapping theorem, thereby obtaining both existence and uniqueness under a natural physical volume bound and providing an explicit iterative scheme for constructing the solution.

The setting is a compact n‑dimensional Riemannian manifold (M,g) with positive Yamabe invariant, no non‑trivial conformal Killing fields, and a prescribed mean curvature τ∈L⁸∩W^{1,n}. A transverse‑traceless (TT) tensor σ∈L² is assumed to be sufficiently small. The conformal data are written as \