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University of Iowa Iowa Research Online Theses and Dissertations Spring 2017 Conformal transformations, curvature, and energy Richard G. Ligo University of Iowa Follow this and additional works at: https://ir.uiowa.edu/etd Part of the Mathematics Commons Copyright © 2017 Richard G. Ligo This dissertation is available at Iowa Research Online: https://ir.uiowa.edu/etd/5550 Recommended Citation Ligo, Richard G.. "Conformal transformations, curvature, and energy." PhD (Doctor of Philosophy) thesis, University of Iowa, 2017. https://doi.org/10.17077/etd.99g6qj3i Follow this and additional works at: https://ir.uiowa.edu/etd Part of the Mathematics Commons

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CONFORMAL TRANSFORMATIONS, CURVATURE, AND ENERGY by Richard G. Ligo A thesis submitted in partial fulﬁllment of the requirements for the Doctor of Philosophy degree in Mathematics in the Graduate College of The University of Iowa May 2017 Thesis Supervisor: Associate Professor Oguz Durumeric

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Copyright by RICHARD G. LIGO 2017 All Rights Reserved

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Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Richard G. Ligo has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Mathematics at the May 2017 graduation. Thesis committee: Oguz Durumeric, Thesis Supervisor Charles Frohman Keiko Kawamuro Walter Seaman Maggy Tomova

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ACKNOWLEDGEMENTS Thank you to my advisor, Oguz Durumeric, a man with whom I have found myself brilliantly compatible in many ways. I am deeply appreciative of his guidance in all avenues—mathematical, professional, and personal. Thank you to all the other mathematicians who have played a role in this journey. To David Stewart: our nine-minute conversation regarding MATLAB was the gateway to nearly a third of my work. To Charlie Frohman: in those times when Oguz and I wanted another opinion, your mathematical wisdom was most useful. To Maggy Tomova: your professional advice has been invaluable, and I credit you with giving me the initiative to work with Oguz. To Keiko Kawamuro and Walter Seaman: I am thankful for your willingness to serve on my review panel alongside Oguz, Charlie, and Maggy. To my colleagues: you have always been willing to listen to my bursts of geometric ideas, despite not being students of the subject. Thank you also to my family. My father’s patient conﬁdence and mother’s unwavering support have served as great motivators. For those times I needed to step away from my work, my brother’s tremendous energy and endless creativity provided welcome respites. Finally, I am thankful to God for having given me this wonderful opportunity and the resilience to bring it to completion. Mathematics will always be one of my favorite parts of His creation. ii

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ABSTRACT 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 Mo¨bius 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 upon those of Freedman, He, and Wang, two topics remain largely unexplored: the interaction of curvature and M¨obius energy and the generalization of the M¨obius 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 2 show that any simple, closed C curve can be transformed in an energy-preserving and length-preserving way that allows us to make the pointwise curvature arbitrarily large 2 at a point. Next, we prove that the total absolute curvature of a C 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 Mo¨bius energy for simple curves of varying thickness that we call the “nonuniform energy.” We call such curves “weighted curves,” and they are deﬁned as the pairing of a curve parametrization and positive, continuous weight function on the same domain. iii

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We then calculate the ﬁrst variation formulas for several diﬀerent variations of the nonuniform energy. Variations preserving a closed curve’s shape and total weight are shown to have no minimizers exactly when the curve has a point of zero curvature. Variations that “slide” the weight along a closed curve are shown to preserve energy is special cases. iv

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PUBLIC ABSTRACT 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 curva- ture. 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. v

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TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Conformal Transformations . . . . . . . . . . . . . . . . . . . . . 1 1.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The M¨obius Energy . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 POINTWISE CURVATURE AND CONFORMAL TRANSFORMA- TIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 TOTAL ABSOLUTE CURVATURE AND CONFORMAL TRANS- FORMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Total Curvature of Polygonal Curves . . . . . . . . . . . . . . . . 25 2 3.2 Total Curvature of C Curves . . . . . . . . . . . . . . . . . . . . 33 3.2.1 Curvature Formulas and Consequences . . . . . . . . . . . 33 2 3.2.2 Analysis of Inversions of Closed C Curves . . . . . . . . 40 3.2.2.1 Inversion Centers on the Curve . . . . . . . . . . 42 3.2.2.2 Inversion Centers oﬀ the Curve . . . . . . . . . . 56 2 3.2.3 Analysis of Inversions of Open, Piecewise C Curves . . . 113 4 THE NONUNIFORM ENERGY . . . . . . . . . . . . . . . . . . . . . 117 4.1 Deﬁnition of the Nonuniform Energy . . . . . . . . . . . . . . . . 117 4.2 Properties of the Nonuniform Energy . . . . . . . . . . . . . . . 118 4.3 Properties of Minimizers of the Nonuniform Energy . . . . . . . 125 4.4 Computing the Nonuniform Energy . . . . . . . . . . . . . . . . 130 5 VARIATIONS OF THE NONUNIFORM ENERGY . . . . . . . . . . 134 5.1 Curve Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.1.1 The Free Curve Variation . . . . . . . . . . . . . . . . . . 138 5.1.2 The Equilength Variation . . . . . . . . . . . . . . . . . . 142 5.2 Weight Variations . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.2.1 The Equitotal Variation . . . . . . . . . . . . . . . . . . . 161 vi

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5.2.2 The Translatory Variation . . . . . . . . . . . . . . . . . 173 5.2.3 The Compromisal Variation . . . . . . . . . . . . . . . . . 193 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 vii

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