MELTING OF THE EUCLIDEAN METRIC TO NEGATIVE SCALAR CURVATURE

We find a C ∞ -continuous path of Riemannian metrics gt on Rk, k � 3, for 0 � t � " for some number " > 0 with the following property: g0 is the Euclidean metric on Rk, the scalar curvatures of gt are strictly decreasing in t in the open unit ball and gt is isometric to the Euclidean metric in the complement of the ball. Furthermore we extend the discussion to the Fubini-Study metric in a similar way.