Date of Degree
PhD (Doctor of Philosophy)
First Committee Member
Second Committee Member
Third Committee Member
Fourth Committee Member
Space curves have a variety of uses within mathematics, and much attention has been paid to calculating quantities related to such objects. The quantities of curvature and energy are of particular interest to us. While the notion of curvature is well-known, the Mobius energy is a much newer concept, having been first defined by Jun O'Hara in the early 1990s. Foundational work on this energy was completed by Freedman, He, and Wang in 1994, with their most important result being the proof of the energy's conformal invariance. While a variety of results have built those of Freedman, He, and Wang, two topics remain largely unexplored: the interaction of curvature and Mobius energy and the generalization of the Mobius energy to curves with a varying thickness. In this thesis, we investigate both of these subjects.
We show two fundamental results related to curvature and energy. First, we show that any simple, closed, twice-differentiable curve can be transformed in an energy-preserving and length-preserving way that allows us to make the pointwise curvature arbitrarily large at a point. Next, we prove that the total absolute curvature of a twice-differentiable curve is uniformly bounded with respect to conformal transformations. This is accomplished mainly via an analytic investigation of the effect of inversions on total absolute curvature.
In the second half of the thesis, we define a generalization of the Mobius energy for simple curves of varying thickness that we call the "nonuniform energy." We call such curves "weighted knots," and they are defined as the pairing of a curve parametrization and positive, continuous weight function on the same domain. We then calculate the first variation formulas for several different variations of the nonuniform energy. Variations preserving the curve shape and total weight are shown to have no minimizers. Variations that "slide" the weight along the curve are shown to preserve energy is special cases.
Curves in n-dimensional space can be described mathematically, which allows us to calculate various quantities related to these objects. The curvature tells us how much a curve is bending, whereas the energy provides a measure of the complexity of the curve. Conformal transformations of n-dimensional space rearrange curves in n-dimensional space in a predictable way, and it can be shown that this rearrangement does not change a curve’s energy. We address two main questions related to the ideas of curvature, energy, and conformal transformations.
First, how are the curvature and energy related to each other? This question ultimately boils down to understanding how conformal transformations affect curvature. We show that while the effects can appear exotic, the total amount of bending can only become so large.
Second, is there a way to describe the energy of a curve with varying thickness? Following this question we investigate how energy varies as the thickness of a curve is adjusted. We show that certain examples behave is nonintuitive ways; for example, the energy of a certain family of curves does not change when the thickness is changed in a specific way.
Conformal transformations, Curvature, Curves, Differential geometry, Energy, Topological geometry
ix, 196 pages
Includes bibliographical references (page 196).
Copyright © 2017 Richard G. Ligo
Ligo, Richard G.. "Conformal transformations, curvature, and energy." PhD (Doctor of Philosophy) thesis, University of Iowa, 2017.