Cross-ratio distortion and Douady–Earle extension: III. How to control the dilatation near the origin

In this paper, we study how the maximal dilatation of the Douady–Earle extension near the origin is controlled by the distortion of the boundary map on finitely many points. Consider the case of points evenly spread on the circle. We show that the maximal dilatation of the extension in a neighborhood of the origin has an upper bound only depending on the cross-ratio distortion of the boundary map on these points if and only if the number n of the points is more than 4. Furthermore, we show that the size of the neighborhood is universal for each n ≥ 5 in the sense that its size only depends on the distortion.