J ul 2 01 2 NORMAL FORM FOR EDGE METRICS

An edge metric is a metric on the interior of a manifold-with-boundary which is singular at the boundary in a manner described by a given fibration of the boundary. The related edge differential operators arise in many settings and have been the subject of much research; see, for example, [Ma]. Our interest in edge metrics arises from the observation that they are a robust class of metrics naturally generalizing the product of a conformally compact metric with a metric on a compact manifold. The AdS/CFT correspondence in physics deals with such product metrics, and the arena of edge metrics appears to be a natural setting for the geometric and analytic questions which arise. The thesis [Ka] considers a problem concerning eleven-dimensional supergravity from this point of view. In this paper we derive a normal form for edge metrics which we expect will be useful in further studies. The normal form is the analogue of geodesic normal coordinates relative to the boundary at infinity. Edge metrics reduce to conformally compact metrics in the special case that the fibers of the boundary are points. Asymptotically hyperbolic (AH) metrics are conformally compact metrics satisfying a particular scalar normalization at infinity. The normal form for AH metrics was derived in [GL] and a different proof was given in [JS2]. This normal form has been useful in a number of problems concerning AH metrics. The existence statement is that if g is AH on X , then there is a diffeomorphism ψ from a neighborhood of {0} × ∂X in [0,∞) × ∂X to a neighborhood of ∂X in X such that ψ|∂X = Id and