# On the proof of the Thin Sandwich Conjecture in arbitrary dimensions

## TL;DR

This paper proves the validity of Wheeler's thin sandwich conjecture in higher dimensions under certain geometric conditions, extending previous 3D results to arbitrary dimensions and establishing the well-posedness of the problem.

## Contribution

It generalizes the thin sandwich conjecture proof to arbitrary dimensions and shows the geometric conditions can always be satisfied, broadening the applicability of the results.

## Key findings

- The thin sandwich problem is well-posed on open sets of initial data in any compact n-dimensional manifold.
- Results extend previous 3D proofs to higher dimensions.
- Geometric hypotheses for the proofs can always be satisfied.

## Abstract

In this paper we show the validity, under certain geometric conditions, of Wheeler's thin sandwich conjecture for higher dimensional theories of gravity. We extend the results shown by R. Bartnik and G. Fodor for the 3-dimensional case in two ways. On the one hand, we show that the results obtained by the mentioned authors are valid in arbitrary dimensions, and on the other hand we show that the geometric hypotheses needed for the proofs can always be satisfied, which constitutes in itself a new result for the 3-dimensional case. In this way, we show that on any compact n-dimensional manifold, n greater or equal to 3, there is an open set in the space of all possible initial data where the thin sandwich problem is well-posed.