Global curvature for surfaces and area minimization under a thickness constraint

Strzelecki, Pawel; von der Mosel, Heiko
April 2006
Calculus of Variations & Partial Differential Equations;Apr2006, Vol. 25 Issue 4, p431
Academic Journal
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 ?[ 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 ?[ X] = ? > 0 provides, naively speaking, the surface with a thickness of magnitude ?; it serves as an excluded volume constraint for X, prevents self-intersections, and implies that the image of X is an embedded C 1-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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