Cohomology for linearized Ricci curvature

Abstract

The Ricci curvature equations are a central subject of study in geometry. However, in the smooth real case, their linear analysis is often confined to settings in which the background metric is Einstein. In this paper, we establish solvability and uniqueness conditions for the linearized problem on any compact Riemannian manifold with boundary. These conditions are formulated in terms of the cohomology of a canonical cochain complex, constructed by means of a generalized Hodge theory based on pseudodifferential methods. An important element of the theory is that it allows the incorporation of tensorial error terms, arising from linearized metric-dependent sources or from connections on the manifold of metrics. Using Bochner technique, we prove vanishing theorems for the cohomology under geometric assumptions on the boundary and error term, without imposing further interior restrictions.