Geometric curvature energies : from local to global ; from discrete to smooth

  • Geometrische Krümmungsenergien : von lokal zu global ; von diskret zu glatt

Scholtes, Sebastian; von der Mosel, Heiko (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2014)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2014

Abstract

In this thesis we consider geometric curvature energies, which are energies defined on curves, or more generally metric spaces, that are in some way connected to curvature. Most of these energies are defined as double or triple integrals. We show that the limit of the inverse circumradius of triangles that converge to a single point in a metric space, called Menger curvature, coincides with the classic curvature of a smooth space curve and that the mere existence of this Menger curvature ensures that a plane continuum already is a smooth curve. Furthermore, we find a way to regard Menger curvature as an embedding curvature and show that there is an analog to Wald's Theorem for this curvature. The supremum of the circumradius on a sphere in a normed vector space, is sufficient to decide wether or not this space can be endowed with an inner product. The integral p-Menger curvature is the triple integral of the p-th power of the inverse circumradius of a metric space. Various closely related energies, so-called intermediate energies, are investigated with respect to topological restrictions, for example the avoidance of ramification points, that the finiteness of the energy imposes on the metric space. Additionally, it is studied which tangential regularity can be expected at every point of an arbitrary subset of R^d if these energies or the integral Menger curvature are finite. We prove the folklore theorem that a hypersurface has positive reach if and only if it is of class C^{1,1}. Besides, we give a new characterisation of sets of positive reach in terms of alternating Steiner formulae for the parallel sets and use these results to shed new light on already known answers to a problem by Hadwiger. In the last part of the thesis we are concerned with discrete versions of integral Menger curvature, thickness and the Möbius energy, which is the double integral over the difference between the squared inverse extrinsic and squared inverse intrinsic distance of all pairs on a space curve. These discrete curvature energies are defined on equilateral polygons of length 1 with n segments. We establish Gamma-convergence results for n converging to infinity. Under additional hypotheses this directly implies the convergence of (almost) discrete minimizers in a fixed knot class to smooth minimizers in the same knot class. Moreover, we show that the unique minimizers of the discrete Möbius energy and discrete thickness amongst all polygons is the regular n-gon.

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