What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?

Basically, I am asking for a deformation of $R^{3}$ which decreases curvature.

**Remarks:** Such a thing wouldn't be possible in $R^{2}$ by the Gauss-Bonnet theorem.

In 3 or more dimensions it is possible by a theorem of Lohkamp stated here (Theorem 1.6).

The same problem, with *positive* instead of *negative* curvature, has no solution by the Positive Energy Theorem (Theorem 1.2 in the link above).