Chord-arc surfaces with small constant. II: Good parametrizations.

*(English)*Zbl 0733.42016Roughly speaking, a chord-arc surface with small constant is a hypersurface in Euclidean space whose unit normal (i.e., the Gauss map) has small BMO norm. To avoid technical problems it is convenient to assume a priori that the given hypersurface is smooth and nice at infinity, and to look for estimates which do not depend quantitatively on these a priori assumptions.

The requirement that the Gauss map have small BMO norm is a scale- invariant “almost flatness” condition that is somewhat subtle because it allows phenomena like spiralling. In a companion paper [same journal 85, No.2, 198-223 (1991; preceding review)] these surfaces were given certain geometrical and analytic characterizations. [See also the author’s paper in Proc. Am. Math. Soc. 112, No.2, 403-412 (1991; last but one review).] The present paper takes up the question of finding good parameterizations of these surfaces. An obvious question is whether these surfaces can be realized as the image of a hyperplane under a bi- Lipschitz mapping with distortion constant close to one. This seems to be hard, and the author conjectures that it is not true in general (except when the surfaces are 1-dimensional, which is trivial). However, certain kinds of fairly well-behaved parameterizations do exist, and the results are best when the surface has dimension 2, thanks to uniformization.

Closely related (and very interesting) results concerning bi-Lipschitz parameterizations of 2-dimensional surfaces that satisfy a stronger condition have been obtained recently by Tatiana Toro.

The requirement that the Gauss map have small BMO norm is a scale- invariant “almost flatness” condition that is somewhat subtle because it allows phenomena like spiralling. In a companion paper [same journal 85, No.2, 198-223 (1991; preceding review)] these surfaces were given certain geometrical and analytic characterizations. [See also the author’s paper in Proc. Am. Math. Soc. 112, No.2, 403-412 (1991; last but one review).] The present paper takes up the question of finding good parameterizations of these surfaces. An obvious question is whether these surfaces can be realized as the image of a hyperplane under a bi- Lipschitz mapping with distortion constant close to one. This seems to be hard, and the author conjectures that it is not true in general (except when the surfaces are 1-dimensional, which is trivial). However, certain kinds of fairly well-behaved parameterizations do exist, and the results are best when the surface has dimension 2, thanks to uniformization.

Closely related (and very interesting) results concerning bi-Lipschitz parameterizations of 2-dimensional surfaces that satisfy a stronger condition have been obtained recently by Tatiana Toro.

Reviewer: St.Semmes

##### MSC:

42B99 | Harmonic analysis in several variables |

30G35 | Functions of hypercomplex variables and generalized variables |

53A05 | Surfaces in Euclidean and related spaces |

##### Keywords:

singular integral operators; Carleson measures; chord-arc surface; hypersurface; BMO norm; good parameterizations; bi-Lipschitz mapping
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##### References:

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