# Reduced thin-sandwich equations on manifolds euclidean at infinity and on closed manifolds: existence and multiplicity

## TL;DR

This paper investigates the well-posedness and solution structure of reduced thin-sandwich equations in general relativity on both asymptotically Euclidean and closed manifolds, revealing conditions for existence and multiplicity of solutions.

## Contribution

It extends analysis of RTSE well-posedness to AE and closed manifolds, identifying conditions for solution parametrization and generic well-posedness.

## Key findings

- RTSE solutions parametrize open subsets of ECE solutions on AE-manifolds with Yamabe positive metrics.
- RTSE are well-posed around umbilical solutions with Killing fields on closed manifolds.
- In the vacuum case on closed manifolds, RTSE are generically well-posed.

## Abstract

The reduced thin-sandwich equations (RTSE) appear within Wheeler's thin-sandwich approach towards the Einstein constraint equations (ECE) of general relativity. It is known that these equations cannot be well-posed in general, but, on closed manifolds, sufficient conditions for well-posedness have been established. In particular, it has been shown that the RTSE are well posed in a neighbourhood of umbilical solutions of the constraint equations without conformal Killing fields. In this paper we will analyse such set of equations on manifolds euclidean at infinity in a neighbourhood of asymptotically euclidean (AE) solutions of the ECE. The main conclusion in this direction is that on AE-manifolds admitting a Yamabe positive metric, the solutions of the RTSE parametrize an open subset in the space of solutions of the ECE. Also, we show that in the case of closed manifolds, these equations are well-posed around umbilical solutions of the ECE admitting Killing fields and present some relevant examples. Finally, it will be shown that in the set of umbilical solutions of the vacuum ECE on closed manifolds, the RTSE are generically well-posed.