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Formulas for Higher Order Loewner Vector Fields A INDEX FORMULAS FOR HIGHER ORDER LOEWNER VECTOR FIELDS A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Steven Edward Broad Frederico Xavier, Director Graduate Program in Mathematics Notre Dame, Indiana April 2009 c Copyright by ! Steven Broad 2009 All Rights Reserved INDEX FORMULAS FOR HIGHER ORDER LOEWNER VECTOR FIELDS Abstract by Steven Edward Broad n Let ∂z¯ be the Cauchy-Riemann operator and f be a C real-valued function in 2 n a neighborhood of 0 in R for which ∂z¯ f = 0 for all z = 0 in that neighborhood. " " Carath´eodory conjectured that a sphere immersed in R3 must have at least two umbilics. This later motivated Loewner to conjecture the bound Ind(∂nf,0) n z¯ ≤ n on the index of vector fields given in complex notation by ∂z¯ f. Both of these conjectures remain open. Recent work of F. Xavier produced a formula for com- puting the index of such vector fields in the case n = 2 using data about the Hessian of f. In this paper, we extend this result and establish an index for- mula for ∂nf that is valid for all n 2. Structurally, our index formula is of the z¯ ≥ form Ind(∂nf,0) = n + (f), where the defect term contains geometric data z¯ Dn Dn extracted from Hessian-like objects associated with higher order derivatives of f. In particular, our index formula computes the Fredholm index of the Toeplitz n operator on the unit circle whose symbol is ∂z¯ f. Dedication To Karen, Emily and Aiden. ii CONTENTS FIGURES . iv SYMBOLS . v PREFACE . vi CHAPTER 1: INTRODUCTION . 1 1.1 An introduction to umbilics . 1 1.2 Conjectures of Carath´eodory and Loewner . 10 1.3 A brief survey of recent work related to umbilics . 11 CHAPTER 2: EQUIVALENCE OF TWO CONJECTURES . 19 n CHAPTER 3: A MATRIX REPRESENTATION FOR ∂z¯ f ........ 30 3.1 Some natural identifications . 30 3.2 Some useful properties of functions in Cn(D,¯ R) . 35 3.3 The matrix representation . 38 n CHAPTER 4: INDEX FORMULAS FOR ∂z¯ f ................ 45 4.1 Background and Bendixson’s index formula . 45 4.2 An index formula for n 2 ..................... 50 4.3 A defect term for Loewner’s≥ conjecture . 57 4.4 Proof of Lemma 4.9 . 66 CHAPTER 5: FURTHER REMARKS . 69 5.1 Application to the Loewner conjecture . 69 5.2 Xavier’s index theorem . 70 5.3 Application to the Carath´eodory conjecture . 71 5.4 Application to Toeplitz operators . 72 BIBLIOGRAPHY . 75 iii FIGURES 1.1 A planar curve showing planar curvature κ 0.33 . 4 ≈− 1.2 Normal curvature. k (p) =1/R .................. 5 | T | 1.3 Triaxial ellipsoid with lines of curvature. 7 1.4 Lemon (D1), Monstar (D2) and Star (D3) configuration. 8 1.5 An embedded sphere with exactly two umbilics. 9 1.6 Lines of curvature configurations for codimension 1 umbilics . 15 4.1 Using Bendixson’s index formula . 49 iv SYMBOLS R The real numbers R2 The real plane C The complex plane D The open unit disk in the complex plane ∂z¯ The partial derivative with respect toz ¯ T The unit circle in the complex plane Tr(A) The trace of a matrix, A (2) The subset of symmetric, S traceless matrices in M2(R) σ A mapping of C into (2) S ι The inclusion of C into R2 Hf The hessian matrix of f (f) The traceless part of H H f n Λn(f) The matrix representation of ∂z¯ f v PREFACE This work represents a significant improvement in the understanding of a con- jecture of Charles Loewner regarding the index of Loewner vector fields at singular points. In the case n = 2, Loewner’s conjecture implies the Carath´eodory conjec- ture about the minimum number of umbilics of immersed spheres. Although this work does not directly improve our collective understanding of the Carath´eodory conjecture, there is hope that it may provide useful notions of higher order um- bilics. More immediately, the proof of a formula for computing the index of a Loewner vector field is broadly applicable in and of itself. Such a formula should provide greater insight into the proof of –or counterexample to – Loewner’s conjecture. Furthermore, it will be useful to mathematicians for whom computing such indices is a necessary (and previously strenuous) task. The following work presents detailed proofs of the main theorems placed in the context of the history of the problem. In Chapter 1, we introduce some of the basic notions related to the study of umbilics and lines of curvature and state the conjectures of Carath´eodory and Loewner. We also discuss some recent progress in problems closely related to this area of study. In Chapter 2, we reproduce a proof of the equivalence of Loewner’s conjecture for n = 2 and the local Carath´eodory conjecture. In Chapter 3, we n develop a matrix representation of the Loewner vector field ∂z¯ f. This is an es- vi sential technical step in approaching the problem of proving an index theorem for Loewner vector fields. In Chapter 4, we use this matrix representation to prove two index formulas for Loewner vector fields. This chapter contains the main result of this paper which provides a defect term for Loewner’s conjecture. In Chapter 5, we discuss some consequences of the index formulas proved in Chapter 4. I would like to thank Fred for being patient and helping me stay focused on the task at hand. He also provided great direction for my work, without which nothing would have happened. Fran¸coisLedrappier helped me broaden my mathematical interests and was always encouraging. I would also like to thank Cabral Balreira whose own experiences going though this process – and his willingness to share them – were a great source of information and reassurance for me. Giuseppe Tinaglia helped me appreciate the fact that everything in mathematics is difficult even if we say otherwise. vii CHAPTER 1 INTRODUCTION The existence of isolated umbilical points on immersed surfaces has been stud- ied since the 18th century. This chapter introduces the reader to two conjectures in differential geometry which are related to the existence of umbilical points, one attributed to Constantin Carath´eodory, the other to Charles Loewner. We will begin with an introduction to umbilics including the definition of an umbilic and some remarks to motivate these conjectures. Once we have made the necessary definitions and some appropriate historical remarks, we will state the conjectures and review their current status. We will conclude with a discussion of precisely how these two conjectures are related to each other. 1.1 An introduction to umbilics In this section, we will review the definitions and basic results related to um- bilical points, or simply umbilics, of a surface immersed in R3. Notions of umbilics exist for higher dimensional immersions. However, the results of this work pri- marily relate to surfaces embedded or immersed in R3. There are several ways to approach the definition of an umbilic. The first notion that we need is that of principal curvature. This requires us to define the first and second fundamental forms of an immersed Riemannian manifold. 1 Definition 1.1. Let x : M R3 be a sufficiently smooth isometric immersion of → the Riemannian manifold M into R3. Given a point p M, the first fundamental ∈ form is defined by I (X ,Y )= X ,Y , p p p ) p p* where Xp,Yp are vectors in TpM. In a (u, v)-coordinate neighborhood of p, we can define the coefficients E = I(xu,xu) 2F = I(xu,xv) G = I(xv,xv) for the first fundamental form. Thus, in this coordinate neighborhood, EF T I(Xp,Yp)=Xp Yp. (1.2) FG Definition 1.3. Let x : M 2 R3 be a sufficiently smooth isometric immersion → of the 2-dimensional Riemannian manifold M into R3. Given a point p M, the ∈ second fundamental form is defined by II p(Xp,Yp)= x (Xp),Yp , )− ∗ * where Xp,Yp are vectors in TpM. The second fundamental form is a bilinear form. Hence we can represent it by a matrix (II ) where II = II (X ,X ) and X ,X forms an orthonormal basis. ij ij i j { i j} The eigenvalues and eigenvectors of (II ij) have a very special meaning, which will 2 be discussed further in what follows. In a (u, v)-coordinate neighborhood of p, we can define the coefficients e = II (xu,xu) 2f = II (xu,xv) g = II (xv,xv) for the second fundamental form. This does not affect the eigenvalues or eigendi- rections in R3 of the second fundamental form, but it does provide a straightfor- ward way of computing quantities related to the second fundamental form as we will see below. Definition 1.4. The eigenvalues k k of II for p M are known as the 1 ≥ 2 p ∈ principal curvatures of M at p. The principal curvatures are important in many undertakings in geometry. Two useful curvature quantities arise from them. The Gauss curvature of M at p 1 is K(p) = det(II p)=k1k2 and the mean curvature of M at p is H(p)= 2 Tr(II p)= 1 2 (k1 + k2). The above definition of principal curvature also allows one to formally define umbilics as follows. Definition 1.5. An umbilic (or umbilical point) of a sufficiently smooth surface immersed in R3 is a point at which the principal curvatures are equal.
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