Recent Progress and Open Problems in Linearization Stability

Summary

INTRODUCTION

Of the many influential contributions made by Dieter Brill to the mathematical development of general relativity, one of particular significance was his discovery together with Stanley Deser, of the linearization stability problem for Einstein's equations [1]. Brill and Deser showed that the Einstein equations are not always linearization stable (in a sense we shall define more precisely below) and they initiated the long (and still continuing) technical program to deal with this problem when it arises.

Our aim in this article is not to review the extensive literature of positive results on linearization stability but rather simply to introduce the reader to this subject and then to discuss some recent research that has developed out of the study of linearization stability problems. These latter include the relationship of linearization stability questions to the problem of the Hamiltonian reduction of Einstein's equations and lead one directly to the study of a number of recent results in pure Riemannian geometry (e.g., the solution of the Yamabe problem by Schoen, Aubin, Trudinger and Yambe, the Gromov-Lawson results on the existence of metrics of positive scalar curvature and the still unfinished classification problem for compact 3-manifolds). They also include a study of the quantum analogue of the linearization stability problem which has been significantly advanced recently by the work of A.