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11 permit one to reduce various constructions and comparisons to finite models of controlled complexity. 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.
29) with respect to suitable convergence of the U ’s. The latter ensures that the limit of the minimizing sequence is actually a minimum. Note that the “obstacle” conditions that U contain the interior of Q0 and be contained in Q1 prevents the minimization from collapsing into something trivial. 29), one cannot expect much in the way of smoothness in general. 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.
19) α(x, 2−j r) ≥ . These j’s represent the “bad” scales for the point x, and below the radius r. 15) implies that Nr (x) < ∞ for almost all x. There is a more quantitative statement which is true, namely that the average of Nr (x) over any ball B in Rd of radius r is finite and uniformly bounded, independently of the ball B and the choice of r. 20) r−d B Nr (x) dx ≤ C(n, −1 f Lip ), where C(n, s) is a constant that depends only on n and s, and f Lip denotes the Lipschitz norm of f . This is a kind of “Carleson measure condition”.