On strong unique continuation of coupled Einstein metrics

TL;DR

This paper proves a strong unique continuation property for Einstein metrics using Carleman inequalities, allowing the result to hold without relying on analyticity and extending to Einstein-scalar-field systems.

Contribution

The paper introduces a new proof technique for unique continuation of Einstein metrics that avoids analyticity by employing Carleman inequalities, applicable to coupled Einstein-scalar-field systems.

Findings

Unique continuation holds for Einstein metrics without analyticity.

Method extends to Einstein-scalar-field systems with cosmological constant.

Provides a robust approach under certain non-analytic perturbations.

Abstract

The strong unique continuation property for Einstein metrics can be concluded from the well-known fact that Einstein metrics are analytic in geodesic normal coordinates. Here we give a proof of the same result that given two Einstein metrics with the same Ricci curvature on a fixed manifold, if they agree to infinite order around a point, then they must coincide, up to a local diffeomorphism, in a neighborhood of the point. The novelty of our method lies in the use of a Carleman inequality and thus circumventing the use of analyticity; thus the method is robust under certain non-analytic perturbations. As an example, we also show the strong unique continuation property for the Riemannian Einstein-scalar-field system with cosmological constant.