On nearly type D solutions of Einstein equation

We study local properties of solutions to 3+1-dimensional Einstein equations which are close to a Schwarzschild metric. Roughly speaking, assume that [Formula: see text] is a local solution to [Formula: see text] in [Formula: see text] and close to a Schwarzschild metric, if its curvature tensor [Formula: see text] is close to a Type D tensor [Formula: see text], then we can find a Kerr metric [Formula: see text] satisfying the interior bound [Formula: see text]