Global curvature for rectifiable loops

Abstract

We study in detail the notion of global curvature defined on rectifiable closed curves, a concept which has been successfully applied in existence and regularity investigations regarding elastic self-contact problems in nonlinear elasticity. A bound on this purely geometric quantity serves as an excluded volume constraint to prevent selfintersections of slender elastic bodies modeled as elastic rods. Moreover, a finite global curvature characterizes simple closed curv es, whose arc length parameterizations possess a Lipschitz continuous tangent field. The investigation of local and non-local properties of global curvature motivates, in particular, an extended definition of local curvature at any point of a rectifiable loop. Finally we show how a bound on global curvature can be used to define and control topological constraints such as a given knot type for closed loops or a prescribed linking number for closed framed curves, suitable to describe, e.g., supercoiling phen omena of biomolecules.