Global curvature for rectifiable loops
Friedemann Schuricht and Heiko von der Mosel
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Submission date: 14. Nov. 2001
published in: Mathematische Zeitschrift, 243 (2003) 1, p. 37-77
DOI number (of the published article): 10.1007/s00209-002-0448-0
MSC-Numbers: 53A04, 57M25, 74K10, 74M15, 92C40
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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 investigation 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 curves, 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 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 phenomena of biomolecules.