Random systems of curves exhibiting fluctuating features on arbitrarily small scales (δ) are often encountered in critical models. For such systems it is shown that scale-invariant bounds on the probabilities of crossing events imply that typically all the realized curves admit Hölder continuous parametrizations with a common exponent and a comon random prefactor, which in the scaling limit (δ ...

We investigate the completeness and completions of the normed algebras (D(1)(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D(1)(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which...

In the present paper we prove that for any open connected set Ω ⊂ R, n ≥ 1, and any E ⊂ ∂Ω with 0 < H(E) < ∞ absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable. CONTENTS

Closed Legendrian (d− 1)-dimensional locally rectifiable currents on the sphere bundle in R are considered and the associated index functions are studied. A topological condition assuring the validity of a local version of the Gauss-Bonnet formula is established. The case of lower-dimensional Lipschitz submanifolds in R and their associated normal cycles is examined in detail.

This paper introduces a notion of decompositions integral varifolds into countably many varifolds, and the existence such decomposition whose first variation is representable by integration established. Furthermore, this result can be generalized replacing class some classes rectifiable density uniformly bounded from below. However, may fail to unique.

Abstract We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. also provide two applications: a Lusin-type approximation ${\mathbb {H}}$-rectifiable sets and coarea formula that completes program started in [18].

We show that a proper metric measure space is a RNP-differentiability space if and only if it is rectifiable in terms of doubling metric measure spaces with some Poincaré inequality. This result characterizes metric measure spaces that can be covered by spaces admitting Poincaré inequalities, as well as metric measure spaces that admit a measurable differentiable structure which permits differe...