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Global curvature for surfaces and area minimization under a thickness constraint

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Abstract

Motivated by previous work on elastic rods with self-contact, involving the concept of the global radius of curvature for curves (as defined by Gonzalez and Maddocks), we define the global radius of curvature Δ[X] for a wide class of continuous parametric surfaces X for which the tangent plane exists on a dense set of parameters. It turns out that in this class of surfaces a positive lower bound Δ[X] ≥ θ > 0 provides, naively speaking, the surface with a thickness of magnitude θ; it serves as an excluded volume constraint for X, prevents self-intersections, and implies that the image of X is an embedded C1-manifold with a Lipschitz continuous normal. We also obtain a convergence and a compactness result for such thick surfaces, and show one possible application to variational problems for embedded objects: the existence of ideal surfaces of fixed genus in each isotopy class.

The proofs are based on a mixture of elementary topological, geometric and analytic arguments, combined with a notion of the reach of a set, introduced by Federer in 1959.

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Authors and Affiliations

  1. Instytut Matematyki, Uniwersytet Warszawski, ul. Banacha 2, 02–097, Warsaw, Poland

    Paweł Strzelecki

  2. Institut für Mathematik, RWTH Aachen, Templergraben 55, 52062, Aachen, Germany

    Heiko von der Mosel

Authors
  1. Paweł Strzelecki
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  2. Heiko von der Mosel
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Correspondence to Heiko von der Mosel.

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Mathematics Subject Classification (2000) 49Q10, 53A05, 53C45, 57R52, 74K15

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Strzelecki, P., Mosel, H.v.d. Global curvature for surfaces and area minimization under a thickness constraint. Calc. Var. 25, 431–467 (2006). https://doi.org/10.1007/s00526-005-0334-9

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