The Cauchy problems for Einstein metrics and parallel spinors

TL;DR

This paper proves local existence of Einstein metrics with prescribed initial data in the analytic setting and addresses the Cauchy problem for metrics with parallel spinors, providing positive results analytically but negative in smooth cases.

Contribution

It establishes local existence results for Einstein metrics and parallel spinors in the analytic category, answering open questions and contrasting with smooth category limitations.

Findings

Existence of Einstein metrics with given initial data in the analytic setting.

Negative results for the smooth category regarding the same problem.

Resolution of a question from B"ar, Gauduchon, Moroianu (2005).

Abstract

We show that in the analytic category, given a Riemannian metric $g$ on a hypersurface $M\subset \Z$ and a symmetric tensor $W$ on $M$, the metric $g$ can be locally extended to a Riemannian Einstein metric on $Z$ with second fundamental form $W$, provided that $g$ and $W$ satisfy the constraints on $M$ imposed by the contracted Codazzi equations. We use this fact to study the Cauchy problem for metrics with parallel spinors in the real analytic category and give an affirmative answer to a question raised in B\"ar, Gauduchon, Moroianu (2005). We also answer negatively the corresponding questions in the smooth category.