Scalar Curvature Decrease From a Hyperbolic Metric

Abstract

We find an explicit C ∞ -continuous path of Riemannian metrics gt on the 4-d hyperbolic space H 4 , for 0 ≤ t ≤ ε for some number ε > 0 with the following property: g0 is the hyperbolic metric on H 4 , the scalar curvatures of gt are strictly decreasing in t in an open ball and g t is isometric to the hyperbolic metric in the complement of the ball.