Exchange lemmas are used in geometric singular perturbation theory to track flows near normally hyperbolic invariant manifolds. We prove a General Exchange Lemma, and show that it implies versions of existing exchange lemmas for rectifiable slow flows and loss-of-stability turning points. © 2007 Elsevier Inc. All rights reserved. MSC: 34E15; 34C30

This paper is devoted to give a simplified proof of the trace theorem for functions of bounded deformation defined on bounded Lipschitz domains of Rn. As a consequence, the existence of one-sided Lebesgue limits on countably H-rectifiable sets is also established.

Any simply connected rectifiable domain Ω can be decomposed into uniformly chord-arc subdomains using only crosscuts of the domain. We show that such a decomposition allows one to construct a map from Ω to the disk which is close to conformal in a uniformly quasiconformal sense. This answers a question of Stephen Vavasis. 1991 Mathematics Subject Classification. Primary: 30C35 Secondary: 30C30,...

We show that if γ is a curve in the unit disk, then arclength on γ is a Carleson measure iff the image of γ has finite length under every conformal map onto a domain with rectifiable boundary. Date: July 27, 2012. 1991 Mathematics Subject Classification. Primary: 30C62 Secondary:

Let E be a set in R with finite n-dimensional Hausdorff measure H such that lim infr→0 r −n H (B(x, r)∩E) > 0 for H-a.e. x ∈ E. In this paper it is shown that E is n-rectifiable if and only if

In this paper a piecewise linear finite element approximation of H-surfaces, or surfaces with constant mean curvature, spanned by a given Jordan curve in R3 is considered. It is proved that the finite element Hsurfaces converge to the exact H-surfaces under the condition that the Jordan curve is rectifiable. Several numerical examples are given.

We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow.

In 1931, Jesse Douglas showed that in Rn, every set of k rectifiable Jordan curves, with k ≥ 2, bounds an area-minimizing minimal surface with prescribed topological type if a “condition of cohesion” is satisfied. In this paper, it is established that under similar conditions, this result can be extended to non-Jordan curves.