Deforming an ε-close-to-hyperbolic metric to a hyperbolic metric

We show how to deform a metric of the form g = g(r) + dr(2) to a metric Hg = Hr + dr(2), which is a hyperbolic metric for r less than some fixed lambda, and coincides with g for r large. Here by hyperbolic metric we mean a metric of constant sectional curvature equal to -1. We study the extent to which Hg is close to hyperbolic everywhere, if we assume g is close to hyperbolic. A precise definition of the close to hyperbolic concept is given. We also deal with a one-parameter version of this problem. The results in this paper are used in the problem of smoothing Charney-Davis strict hyperbolizations.