ON A MATHEMATICAL MODEL FOR THICK SURFACES
Paweł Strzelecki, Heiko von der Mosel · Series on knots and everything · 2005
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 4[X] for a wide class of parametric surfaces. It turns out that a positive lower bound 4[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. Taking into account possible applications to variational problems for embedded objects, we also obtain a convergence and a compactness result for such thick surfaces. The main object of this note is to introduce and explain the crucial notions and results. Thus, we defer almost all proofs to our forthcoming paper.22 1.