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Reverse Isoperimetric Inequalities in R3 REVERSE ISOPERIMETRIC INEQUALITIES IN R3 DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Andrew Gard, BS Graduate Program in Mathematics The Ohio State University 2012 Dissertation Committee: Dr. Fangyang Zheng, Advisor Dr. Ulrich Gerlach Dr. Bo Guan Dr. Chris Hans c Copyright by Andrew Gard 2012 ABSTRACT We formulate conditions under which the classical isoperimetric inequality in R3 can be reversed. Restricting our attention to surfaces with rotational symmetry, we enforce bounds on curvature and overall size (both in the appropriate technical senses) and show that these suffice to guarantee the existence of shapes of minimal volume for given fixed surface area. In the fundamental case where both principal curvatures are bounded, we construct the surface of minimal volume. ii For my family. iii ACKNOWLEDGMENTS The first thanks of course go to my advisor, Dr. Fangyang Zheng, who brought the seed of this project to me and suggested how I could make it grow. His support and advice have been invaluable throughout this process. I also wish to thank professors Chris Hans, Bo Guan and Ulrich Gerlach for generously taking the time to review this document and serve at my defense. I recognize that this is not a small commitment; all I can do in return is to attempt to be worthy of their efforts. Without the support of friends and family, this project would have driven me mad. Thanks particularly go to Isabel and Jenny for the coffee, Moy and Justin for the debates, and Bob and Billy for the nights off. Thanks also to the denizens of MW200 past and present for the never-ending sharing of mathematical ideas. We'll tease out the truth yet. Finally and most importantly, my wife Kristina, too-often a math widow of late, has my infinite gratitude and love, as does my son Clayton, who still thinks that math just means adding things up. You may be right after all, son. iv VITA 1974 . Born in Hammond Indiana, USA 1999 . B.Sc. in Mathematics, The Ohio State University 2004-Present . Graduate Teaching Associate, The Ohio State University FIELDS OF STUDY Major Field: Mathematics Specialization: Differential Geometry v TABLE OF CONTENTS Abstract . ii Dedication . iii Acknowledgments . iv Vita......................................... v List of Figures . vii CHAPTER PAGE 1 Introduction . 1 1.1 A Little History . 1 1.2 Preliminaries . 3 2 Existence: jκθj; jκσj ≤ 1........................... 6 2.1 Formulation of the Problem . 6 2.2 Existence of the Minimal Shape . 15 3 Construction of the Minimal Shape: jκθj; jκσj ≤ 1 . 17 4 Existence: jHj ≤ 1 ............................. 40 4.1 Formulation of the problem . 40 4.2 Existence of the Minimal Shape . 55 Bibliography . 58 vi LIST OF FIGURES FIGURE PAGE 1.1 The Howard/Treibergs result . 2 2.1 A typical element of FA0 ......................... 7 2.2 The threshold σT ............................. 8 2.3 From axis to cap . 12 3.1 The neighborhood to be deleted . 28 3.2 The optimal σmin ............................. 31 3.3 Swapping pinwheels while preserving ∆x . 32 3.4 Connecting a local minimum . 33 3.5 Minimal area when β = π=2 ...................... 34 3.6 Yet another simplifiction . 35 3.7 The last simplification . 37 3.8 Nearing a hollow sphere . 39 3.9 Adding area without adding volume . 39 4.1 Cap comparison . 46 4.2 A typical jHj ≡ 1 sleeve . 50 4.3 A typical jHj ≡ 1 half-balloon . 53 vii CHAPTER 1 INTRODUCTION 1.1 A Little History The isoperimetric inequality says, roughly, that among domains of of fixed boundary area the one having greatest volume is the n-sphere. Analytically, 1−1=n A(@Ω) V (Ω) ≤ 1=n (1.1.1) n!n n Where Ω is any domain in R , V is n-dimensional volume, A is (n − 1)-dimensional n boundary area, and !n is the volume of the unit ball B . Equality holds exactly when Ω = Bn. In the plane, this result was known to the ancient Greeks and proved by Steiner in 1838. Countless new proofs have emerged over the years since, drawing on nearly every branch of mathematics and extending the result in every conceivable direction, most notably to manifolds and abstract measure spaces. See [3] for a somewhat recent survey. The opposite inequality remains almost completely unaddressed. In a sense this is logical. There is in general no minimum volume for fixed surface area (or maximum surface area for fixed volume). Restrictions must be imposed before the question becomes meaningful. Yet the conditions that we must set turn out to be so simple and obvious - bounds on curvature and overall size - that there need be no unbearable loss of elegance in the statement of the problem. 1 Moreover, the question is not without physical interpretation or application. One needs to look no further than the dimple on the bottom of a bottle of wine to see the commercial desirability of volume reduction. In the human body, the red blood cell carries oxygen on its exterior, and so naturally seems to desire greater surface area for relatively fixed volume. In a sense, nature has already addressed this problem. In 1995, Howard and Treibergs [8] proved a reverse isoperimetric inequality for simple plane curves, finding for perimeter 2π ≤ L < 14π=3 and curvature jκj ≤ 1 the relation L − 2π A ≥ π + 4 sin (1.1.2) 4 where L represents perimeter length. The extremal shape resembles a peanut: Figure 1.1: The Howard/Treibergs result As L increases, the concave portions of the peanut pinch inward until finally meeting when L = 14π=3. The reverse-isoperimetric problem ceases to be meaningful beyond this threshhold. p Figure 1.1 is also extremal for the dual problem, in which A 2 [π; π + 2 3] is p fixed and L is maximized. Here the threshhold value A = π + 2 3 corresponds to the situation where L can be made arbitrarily large by pinching the concave parts of the peanut and stretching the convex parts away from one another like taffy. 2 In higher dimensions the reverse isoperimetric problem rapidly becomes more difficult, with multiple formulations possible. In this dissertation, we address the rotationally-symmetric three-dimensional case, showing that the problem is well- defined when either mean curvature or both principal curvatures are bounded and surface area A0 is less than a determined threshhold. In the principal curvature case, we construct the optimal shape. 1.2 Preliminaries Most of this dissertation will be derived from first principles, but we will occasionally require some fundamental ideas from local theory of smooth manifolds. Here we establish those results. 1 2 3 Let X be a differentiable map from a domain D in the (x ; x )-plane into R . By considering the partial derivatives of X we obtain the tangent space TpX = SpanfX1;X2g of X at point p and the first fundamental form I : TpX × TpX ! R of the surface X(D): 0 1 hX ;X i hX ;X i B 1 1 1 2 C (gij) = @ A hX2;X1i hX2;X2i 3 where the inner product is inherited from R . This matrix is non-singular whenever X1 and X2 are linearly independent (so-called regular points). For convenience we set ij −1 (g ) = (gij) and g = det(gij). This Riemannian metric is the basis for all intrinsic calculations of length, angle, and area. Since all the surfaces in this dissertation will be C1 or better, the reliability of this metric will never be at issue. X ^ X The unit normal n = 1 2 is now well-defined and independent of our choice jX1 ^ X2j coordinates (x1; x2) up to sign. The second fundamental form II is then given in the 3 basis fX1;X2g by: 0 1 0 1 hX ; ni hX ; ni −hX ; n i −hX ; n i B 11 12 C B 1 1 1 2 C (hij) = @ A = @ A (1.2.1) hX21; ni hX22; ni −hX2; n1i −hX2; n2i Closely related to II is the shape operator S (or Weingarten map) on TpX, defined by S(u) = −Dun. Like the second fundamental form, the it tracks changes in the unit 3 normal and thus gives information about the ways in which X(D) curves locally in R . The shape operator satisfies the important properties II(u; v) = hS(u); vi = hu; S(v)i. The first equality is a matter of definition; let us verify the second. j Since hXi; ni = 0, we have hXi; nji = −hn;Xiji = hXj; nii. Writing u = u Xj and j v = v Xj, where summation is understood to take place whenever an index occurs in both a lower and upper position, we obtain: i j i j hS(u); vi = −hu ni; v Xji = −u v hni;Xji (1.2.2) i j i j = −u v hXi; nji = −hu Xi; v nji = hu; S(v)i (1.2.3) In light of the fact that 0 = Dwhn; ni = 2hDwn; ni, S is a map from TpX to itself j and we may write S(Xi) = Si Xj. Direct computation shows that in fact: j X kj Si = hkig (1.2.4) X jk ni = − hijg Xk (1.2.5) The eigenvectors of S represent the directions of minimal and maximal changing of n. The corresponding eigenvalues are called the principal curvatures of X(D). The de- terminant and trace of S are called the Gauss and mean curvatures, respectively. The former can be shown to be depend only on (gij) and its derivative (Gauss's Theorema egregium), and hence measures the way in whichthe surface curves intrinsically.
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