By S. Semmes
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29) for a suitable choice of g. Specifically, one can take g to be a sufficiently small positive constant on ∂U , and to be equal to 1 everywhere else. That such a choice of g works is not very hard to establish, and more precise results are given in [DaviS9]. 29) are always Ahlfors-regular sets of dimension n − 1, and uniformly rectifiable. 29) (with g bounded and bounded away from 0). , a small positive constant on ∂U and equal to 1 everywhere else. 29) as a special case. Uniform rectifiability provides a natural level of structure for situations like this, where stronger forms of smoothness cannot be expected, but quantitative bounds are reasonable to seek.
8), (2) families of curves in M which are well-distributed in terms of arclength measure, and (3) mappings to spheres with certain estimates and nondegeneracy properties. These three kinds of information are closely linked, through various dualities, but to some extent they also have their own lives. Each would be immediate if M had a bilipschitz parameterization by Rn , but in fact they are more robust than that, and much easier to verify. Indeed, one of the original motivations for [DaviS1] was the problem of determining which conformal deformations of Rn lead to metric spaces (through the geodesic distance) which are bilipschitz equivalent to Rn .
One does have mass bounds for the examples in [DaviS3] (of totally-unrectifiable Ahlfors-regular sets for which the β(x, t)’s tend to 0 uniformly as t → 0), and there the issue is more in the size of the holes in the set. The bβ’s, by definition, control the sizes of holes. Note that this result for the bβ’s does have antecedents for the classical notion of (countable) rectifiability, as in [Mat]. There are a number of variants of the bβ’s, in which one makes comparisons with other collections of sets besides d-planes, like unions of d-planes, for instance.