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.
If p M is an umbilic, the definitions above allow one to conclude that II = λI ∈ p for some λ R. This property is quite useful, but it is not yet entirely clear what ∈ other properties an umbilic might have. An equivalent definition of the principal curvatures can give us a little more insight into their properties:
3 T' t
c t ! "
! " T t
N t ! "
! "
Figure 1.1. A planar curve showing planar curvature κ 0.33 ≈−
Definition 1.6. If c :(a, b) R2 is a C2 regular curve in the plane, then the → tangent and normal vectors are defined at t (a, b) to be ∈
c (t) 0 1 T&(t)= " and N& (t)= − T&(t). c (t) + " + 10 In other words, the normal vector is the tangent vector rotated counterclock- π & & wise by 2 . As such, the pair of vectors (T (t), N(t)) forms a positive basis.
Definition 1.7. If c :(a, b) R2 is a C2 parametrized curve in the plane, then → the planar curvature of c at t is defined to be
T& "(t)=κc(t)N& (t).
According to this definition, κ(t) could be negative as in the scenario shown in Figure 1.1. The next step is to define a notion of curvature on a surface that
4 N N
T T p R
T'
Figure 1.2. Normal curvature. k (p) =1/R | T |
uses this notion of planar curvature.
Definition 1.8. Suppose x : M R3 is an immersion which defines a surface. → Then for all p in M, one can choose a vector N in R3 which is normal to the plane tangent to x(M) at x(p). If ν is a unit tangent vector to x(M) at x(p), then N and ν define a plane which cuts through the surface x(M) in a curve C. The curve
C is a planar curve, and therefore it can be parametrized by c :( ε,ε ) R3, − → where c(0) = x(p) and c"(0) = ν. Then the normal curvature in the direction of ν at p is defined to be
kν(p)=κc(0).
The definition of normal curvature is illustrated in Figure 1.2. The image on the left shows a surface with normal vector N at a point p. A vector T tangent to the surface at p is chosen and the plane through N and T is shown. The intersection of the plane and the surface forms a curve. This curve is shown
5 projected into R2 on the right. The circle shown is the largest circle which is tangent to the curve at p on the side of T "(p) which is on one side of the curve in a neighborhood of p in the plane. The definition of normal curvature gives rise to another equivalent definition of principal curvature.
Proposition 1.9. The principal curvatures, k1,k2, at a point p in M are defined by
k1(p) = max(kν(p)) and k2(p) = min (kν(p)). ν =1 ν =1 # # # # The fact that this statement is equivalent to Definition 1.4 is well-known and proved in many textbooks, for example [6]. Property 1.9 together with Definition 1.5 means that at an umbilic point the normal curvature in every direction is the same. This is the case at every point on the sphere. Hence, every point on the sphere is an umbilic. For this reason, umbilics are sometimes called locally spherical points. The principal curvatures of M at p are scalar quantities, but since they are eigenvalues of (II ij), they are associated with eigenvectors. The eigenvectors asso- ciated with k1 and k2 are exactly the choices of ν in Property 1.9 which respectively maximize and minimize the normal curvature. Since (II ij) is symmetric as long as the point p in question is not an umbilic (i.e. k1 >k2), the eigenvectors of (II ij) are orthogonal. Therefore, trajectories through the field of eigendirections corre- sponding to k1 and k2 each form the leaves of a foliation of M. These foliations intersect transversally – in fact, orthogonally – at every non-umbilical point.
In classical studies of umbilics by Monge [26], Darboux [5] and others, these trajectories were studied as integral curves. In particular, these curves can be
6 Figure 1.3. Triaxial ellipsoid with lines of curvature.
obtained by integrating the equation
du2 du dv dv2 − ' ' ' EFG ' =0, (1.10) ' ' ' ' ' ' ' e f g ' ' ' ' ' ' ' where u and v are local coordinates' for M and E,' F, G, e, f, g are the coefficients of the first and second fundamental forms as described above. Solving this equation yields the eigendirections corresponding to k1 and k2. Integrating these eigendi- rections yields a curve which is a trajectory through this field. Since these integral curves follow the field of principal directions, they are called the lines of curvature. An excellent discussion of this classical approach is given in Sections 2-6 through 2-9 in the classic text by Dirk Jan Struik, Lectures on
Classical Differential Geometry [36]. The lines of curvature of the triaxial ellipsoid were computed by Gaspard Monge in the last decade of the 18th century [32]. The
7 D1 D2 D3
Figure 1.4. Lemon (D1), Monstar (D2) and Star (D3) configuration. triaxial ellipsoid is – up to rigid motion – the surface of solutions to the equation
x2 y2 z2 + + = 1 (1.11) a2 b2 c2 where a, b, c are three distinct positive real numbers. In Figure 1.3, one can see that the principal lines collapse to four separate line segments whose union is an equator of the ellipsoid. The end points of these four line segments – which are marked on the surface – are the umbilics of the ellipsoid, since the eigenvectors corresponding to both eigenvalues are parallel, meaning that the principal curvatures must be equal. In the last decade of the 19th century, Jean Gaston Darboux classified the pos- sible configurations of lines of curvature in the neighborhood of a generic umbilic.
There are three, namely the lemon (D1) and star (D3) configurations and the monstar (D2) configuration which is a blending of lemon and star as shown in Figure 1.4. One can easily see that the four umbilics of the triaxial ellipsoid are all in the lemon configuration.
Consider the ellipsoid obtained by revolving an ellipse around either its major or minor axis. This corresponds to allowing a = b in (1.11). It is also worth
8 Figure 1.5. An embedded sphere with exactly two umbilics.
noting that this surface has exactly two umbilics, instead of four as on the tri- axial ellipsoid. Furthermore, the tighter lines of curvature (the circles centered along the long axis of the ellipse) collapse to a point. However, we know from our discussion of the triaxial ellipse that if we perturb either a or b in (1.11), the umbilic at each end will split into two umbilics. In this sense, the ellipse of revolution is a non-generic surface. This explains its failure to belong to one of the three generic configurations described by Darboux. Breaking the axial symmetry of the ellipsoid of revolution is also precisely the reason that the two umbilics split into four. A more recent result gives a geometric and structural condition that explains why the transition from ellipsoid of revolution to triaxial ellipsoid has this effect on the umbilics. Theorem 2 in [29] states that if the critical sets of the mean curvature and Gauss curvature of a compact orientable surface M immersed in R3 are disjoint – as is the case for the
9 triaxial ellipsoid, but not for the ellipsoid of revolution – then the surface has at least 2 χ(M) umbilics, where χ(M) is the Euler number of M. Since χ(S2) = 2, | | the triaxial ellipsoid must have at least 4 umbilics. There is much more that could be said about the theory of umbilical points. We will, however, restrict ourselves to the above introduction.
1.2 Conjectures of Carath´eodory and Loewner
It is a curiosity of the study of umbilics that no C2-embedded sphere having exactly one umbilic is known. It is certain that such a surface must have at least one umbilic, as shown in the following proposition. Indeed, this fact is an immediate corollary of the Poincar´e-Hopfindex theorem for line fields, which can be found in [22, p.107-118].
Proposition 1.12. If x : S2 R3 is a C2-embedding, then x(S2) has at least → one umbilic.
The sharpness of the above proposition has long been debated. No examples of immersed spheres with exactly one umbilic have arisen. This surprising absence motivated the following conjecture. It is unclear when, how and in what context it was proposed, yet it was attributed to Carath´eodory by Hamburger in 1924 [20].
Conjecture (Carath´eodory). Every convex C2 embedding of S2 into R3 has at least two umbilic points.
Many proofs of the Carath´eodory conjecture have been attempted, for exam- ple [4],[19],[23]. Each of the cited proofs have had flaws [37]. Some more recent attempts (e.g. [37]) have attempted to prove the Carath´eodory conjecture (which, to the best of our knowledge, remains open) by proving a stronger conjecture of
10 n Charles Loewner about vector fields of the form ∂z¯ f with 2∂z¯ = ∂x + i∂y, where f is a real-valued function. The following conjecture is attributed to Charles Loewner. The historical significance of this conjecture is its relationship to the study of umbilic points of surfaces. Specifically, the Loewner conjecture when n = 2 implies the long- standing conjecture of Carath´eodory stated above.
n ¯ n Conjecture (Loewner (c. 1950)). If f C (D, R) and ∂z¯ f =0for z =0, then ∈ " " n the index at zero of the vector field given in complex form by ∂z¯ f is at most n.
There have been recent attempts to prove Loewner and Caratheodory. How- ever, much of the current research has focused on refining our collective under- standing of these conjectures from new points of view. For example, [16] and [30] exploit connections with dynamical systems and hyperbolic partial differential equations respectively. The main theorem of [39] is an index formula for Loewner
2 vector fields in the case n = 2 which relates the index of ∂z¯f to eigenvalues and eigenvectors of the Hessian of f on the unit circle.
1.3 A brief survey of recent work related to umbilics
A very nice survey article on this topic was written from the point of view of lines of curvature by Garcia and Sotomayor [10, 33]. What follows is not intended to be thorough, but rather to give some flavor of the recent work in this area. The respective works of Carlos Gutierrez, Ronaldo Garcia, Federico S´anchez-Bringas,
Jorge Sotomayor, Frederico Xavier, Bryan Smyth are characteristic of this program
[12–14, 16, 28–31, 35, 39].
11 1.3.1 Structural stability
The work of Andronov-Pontrjagin [1] established the foundation for study of the structural stability of differential equations. Peixoto [27] extended the notion of structural stability to flows on surfaces. This development has had a deep impact on the study of umbilics. We will consider a specific subset of the set of smooth compact oriented surfaces in which its power is evident.
Definition 1.13. LetΣ( a, b, c, d) be the set of smooth compact oriented surfaces S satisfying the following conditions:
(a) All umbilic points are Darbouxian. In other words, the lines of curvature in a neighborhood of each umbilic conforms to one of the three Darboux configurations shown in Figure 1.4.
(b) All principal cycles are hyperbolic. Note that a principal cycle γ is hyperbolic if and only if dH =0, 2 γ √H K " ( − 1 where H = 2 (k1 + k2) is the mean curvature and K = k1k2 is the Gauss curvature.
(c) The limit set of every principal line is contained in the set of umbilics together with the principal cycles of S.
(d) All umbilic separatrices are separatrices of a single umbilic point. In particu-
lar, there are no umbilic connections.
r A sequence Sn of surfaces are said to converge in the C sense to a surface S if there is a sequence of functions fn : S R such that Sn =(I + fnN)(S) and → r fn tends to 0 in the C topology, where N is the unit normal to the surface S.
12 The notions of Cr-open, -closed and -dense sets of surfaces Σ are determined by extending the corresponding definitions for fn. A surface S is said to be Cr-principal structurally stable if for every sequence S converging to S in the Cr sense, there is a sequence H : S S of homeo- n n n → morphisms converging to the identity on S which satisfy the following conditions for large enough n:
(a) Hn maps the umbilics of Sn onto the umbilics of S.
(b) Hn maps the principal foliations of Sn onto the principal foliations of S.
Theorem (Structural Stability of Principal Congurations [34]). The set of sur- faces Σ(a, b, c, d) is open in the C3 sense and each of its elements is C3-principal structurally stable.
Theorem (Density of Principal Structurally Stable Surfaces [15]). The set Σ(a, b, c, d) is dense in the C2 sense.
1.3.2 Lines of curvature
As noted above, the isolated umbilics are singular points of the principal con- figuration of an immersed surface. The principal configuration consists of two orthogonal foliations determined by the lines of principal curvature with the um- bilic points removed and the set of umbilic points. While the behavior of lines of curvature near umbilics is interesting in many ways, some global behaviors and behaviors local to other structures bear consideration.
1.3.2.1 Non-darbouxian umbilics
The method of analysis that categorizes the three Darbouxian configurations proceeds by considering the surface to be locally represented as the graph of some
13 function f : U R which vanishes at the umbilic over its tangent plane. Thus U → is a region in the tangent plane to the surface at the umbilic p and (u, v, f(u, v)) coincides with the surface for (u, v) R2. One then analyzes the function f in ∈ terms of the degree 3 terms of its Taylor polynomial (after some rotation in the (u, v)-plane to eliminate the u2v term):
k a b c (u2 + v2)+ u3 + uv2 + v3 + O[(u2 + v2)]. 2 6 2 6
In the Darbouxian case, the following conditions hold:
T) b(b a) =0 − "
D) The ratio a/b falls in one of the following intervals:
D : ((c/2b)2 +2, ) 1 ∞ D : (1, (c/2b)2 + 2), and a =2b 2 " D :( , 1) 3 −∞
In the above conditions, cases D1,D2 and D3 correspond to the Darboux con- figurations of the same names (Figure 1.4). However, if either condition (T) or condition (D) is violated, the umbilic is not Darbouxian. Specifically, we will ex- amine the notion of isolated umbilics which are said to be of codimension one [17].
The codimension of an isolated umbilic is the minimal number of (bifurcation) parameters that parametrize the families of immersions which retain the same structure local to its umbilics. Umbilics of codimension one and two have been categorized, but we will focus on codimension one.
Umbilics of codimension one fail to satisfy the conditions for Darbouxian um- bilics in the simplest possible ways. These umbilics occur in two configurations:
14 1 1 D2 D2,3
Figure 1.6. Lines of curvature configurations for codimension 1 umbilics
1 1 D2,3 and D2. The codimension of the of umbilic is indicated by the superscript and – as with the Darbouxian umbilics – the subscript indicates the number of separatrices of the field of principal lines that approach the umbilic. In the case of
1 D2,3, the minimal and maximal line fields have different numbers of separatrices. Our characterization of Darbouxian umbilics required some analysis of the 3- jet of the function whose graph over the tangent plane at the umbilic coincides locally with the surface. In the case of umbilics of codimension one (resp. two), one must similarly analyze the 4-jet (resp. 5-jet) of the same function, namely:
k a b c f(u, v)= (u2 + v2)+ u3 + uv2 + v3 2 6 2 6 α β γ δ ε + u4 + u3v + u2v2 + uv3 + v4 + h.o.t. (*) 24 6 4 6 24
Theorem ([17]). Let p0 be an umbilical point and consider α(u, v)=(u, v, f(u, v)) as in equation (*). Suppose the following conditions hold:
D1) cb(b a) =0and either (c/2b)2 (a/b) + 1 = 0 or a =2b. 2 − " −
D1 ) b = a =0and χ = cβ (γ α +2k3)b =0. 2,3 " − − "
15 1 Then the behavior of lines of curvature near the umbilical point p0 in cases D2
1 and D2,3 is as illustrated in Figure 1.6.
A similar theorem is presented in [17] for umbilics of codimension two.
1.3.2.2 Dense principal lines
Historically, the existence of dense leaves of foliations has been studied in a variety of settings. The question of whether or not principal foliations can have dense leaves has recently been answered. Garcia and Sotomayor [11] proved that the 2-torus can be embedded in R3 in such a way that the leaves of the principal foliations are asymptotically dense.
Theorem (Garcia-Sotomayor [17]). There exist embeddings α : T2 S3 such → 2 that all leaves of both asymptotic foliations, α,1 and α,2 are dense in T . A A
After stereographic projection from S3 into R3, the question is answered. The proof of this theorem relies on a second order variation on the usual embedding of the Clifford torus into S3.
1.3.3 Surfaces immersed in R4
Suppose α : M 2 R4 is an immersion of a surface into R4. The first and → second fundamental form of such a surface in local coordinates are given by
I = Dα,Dα = Edu2 +2F du dv + Gdv2 α ) * II = N ,D2α N + N ,D2α N α ) 1 * 1 ) 2 * 2 2 2 2 2 =(e1 du +2f1 du dv + g1 dv )N1 +(e2 du +2f2 du dv + g2 dv )N2
16 where E = α ,α , F = α ,α and G = α ,α . One may then define the ) u u* ) u v* ) v v* ellipse of curvature to be
= II (X,X):X T 1M , Eα { α ∈ }
where II α takes values in the normal bundle of M and X is a section of TM. One says that p is an axiumbilic singularity if the ellipse of curvature at p degenerates into a circle. Similar to surfaces immersed in R3 there are notions of structural stability associated with surfaces immersed in R4 a categorization of the codimension 0 case as with Darbouxian umbilics. However, it takes two flavors called principal axial stability and mean axial stability. For each of these flavors, there is a class of immersions similar toΣ( a, b, c, d). These are denoted k P and k (where k denotes that this space is a subspace of the space of Ck(M 2, R4) Q immersions) for the principal and mean cases respectively. The following theorem is analogous to the structural stability theorem from Section 1.3.1.
Theorem ([9]). Let k 5. The following holds: ≥
1. k and k are open in the space of Ck immersions of M into R4; P Q
2. Every α k is principal axial stable; ∈P
3. Every α k is mean axial stable. ∈Q
In higher dimensions, J. A. Little [24] studied structures analogous to the el- lipse of curvature in the case where α : M n RN . In such cases, one investigates → 1 n 1 the image of the second fundamental form restricted to T M ∼= S − . These images are in fact Veronese manifolds. Little also noted that there are some
17 likely connections between Loewner’s conjecture and certain Veronese manifolds in higher dimensional cases [25]. In [38], C.T.C. Wall remarks on the possibility of further generalizing Little’s notions to create novel classes of geometric singu- larities. Another view one may adopt of the principal configurations of α is to choose a smooth section of the normal bundle of M. This is precisely the approach X taken in [14] where it is shown that a local Loewner-type conjecture does not hold in the n = 2 case.
Theorem ([14]). Given n Z, p R2, there exists a local analytic immersion ∈ ∈ f :(V, p) R2 R4, where V R2 is an open neighborhood of p and is real ⊂ → ⊂ X analytic vector field normal to f(M) such that f(p) is an isolated -umbilic of X index n/2.
18 CHAPTER 2
EQUIVALENCE OF TWO CONJECTURES
In this chapter, we describe one illuminating version of the argument that Loewner’s conjecture when n = 2 implies the Carath´eodory conjecture. The argument proceeds by way of hyperbolic partial differential equations and is taken
2 from “Real solvability of the equation ∂z¯ω = ρg and the topology of isolated umbilics” by B. Smyth and F. Xavier [30].
Theorem. If Loewner’s conjecture holds for n =2, then the Carath´eodory con- jecture also holds.
The proof of this theorem proceeds through the Local Carath´eodory conjecture which is a stronger version of the Carath´eodory conjecture.
Conjecture (Local Carath´eodory Conjecture). An isolated umbilic on a smooth surface in R3 has index 1. ≤ This local version of the Carath´eodory conjecture is introduced in [18]. The fact that Local Carath´eodory implies Carath´eodory is easy to see by noting that if an immersed sphere had exactly one umbilic, the index of that umbilic would have to be 2 matching the Euler characteristic of the sphere. In the remainder of this focus on proving that Local Carath´eodory and Loewner with n = 2 are equivalent, which is sufficient to prove the above theorem. The proof in [30] relies on the notion of Hessian foliations.
19 Definition 2.1. A smooth one-dimensional foliation of a neighborhood Ωof o F ∈ R2 with an isolated essential singularity at o is called a singular Hessian foliation if there exists a smooth real-valued function ω on Ω whose Hessian operator
ωxx ωxy Hω = ωxy ωyy has the following properties:
(a) H is not a multiple of the identity for any p Ω o . ω ∈ −{ }
(b) The eigenspace corresponding to the large (or small) eigenvalue of Hω is tan- gent to for each p Ω o . F ∈ −{ }
Suppose that is a one-dimensional foliation ofΩ o and that the leaves of F 0 are integral curves of the unit vector field represented by the complex-valued F function ζ :Ω T = z =1 C. Then is a singular Hessian foliation if → {| | }⊂ F and only if there exists ω :Ω R such that →
H ζ,Jζ 0 and H = λI onΩ o , (2.2) ) ω *≡ ω " −{ } where J is the matrix that corresponds to multiplication by i in the complex plane. (In the forward direction at each point ofΩ o , ζ corresponds to an eigenvector −{ } of Hω and the eigendirections are orthogonal at each such point. In the reverse direction, the identity implies that ζ is an eigenvector at each point.)
We will find an alternative form of (2.2) to be more helpful.
Proposition 2.3. Given a unit vector field ζ on Ω o , the following are equiv- −{ } alent:
20 (a) The foliation represented by ζ is a singular Hessian foliation of Ω. F
+ (b) There exist ω C∞(Ω o , R) and ρ C∞(Ω o , R ) C(Ω, R) such ∈ −{ } ∈ −{ } ∩ that ∂2ω = ρζ2 on Ω o . z¯ −{ }
Proof. The computation is relatively straightforward, but is included for com- pleteness. Let ζ = a + ib, and recall that ∂2 =(∂ + i∂ )2 = ∂ ∂ +2i∂ . z¯ x y xx − yy xy
ωxx ωxy a b H ζ,Jζ = , − ) ω * ) ωxy ωyy b a * aω + bω b xx xy − = , ) aωxy + bωyy a * = ab(ω ω ) (b2 a2)ω − xx − yy − − xy =(a2 b2)ω ab(ω ω ). − xy − xx − yy
Furthermore,
Im((∂2ω)ζ¯2) = Im((ω ω +2iω )(a2 b2 2iab)) z¯ xx − yy xy − − = Im((ω ω )(a2 b2)+4abω ) xx − yy − xy + Im(2i((a2 b2)ω ab(ω ω ))) − xy − xx − yy = 2(a2 b2)ω 2ab(ω ω ) − xy − xx − yy =2 H ζ,Jζ . ) ω *
Thus the condition for ζ to represent a singular Hessian foliation is the existence
21 of ω :Ω R such that →
Im((∂2ω)ζ¯2) = 0 and ∂2ω = 0 onΩ o . z¯ z¯ " −{ }
2 ¯2 2 ¯2 + If Im((∂z¯ω)ζ ) 0 then (∂z¯ω)ζ = ρ, where ρ :Ω o R . Hence, ≡ −{ }→ 2 2 2 ∂z¯ω = ρζ . Since o is an essential singularity of ∂z¯ω, the continuous extension of ρ to Ωis ρ(o) = 0.
One obvious consequence of this proposition is a relationship between the index
2 of the singular Hessian foliation and the index of the vector field ∂z¯ω:
2 2 2 Ind(∂z¯ω, o) = Ind(ρζ ,o) = Ind(ζ ,o) = 2Ind(ζ, o). (2.4)
Having introduced the notion of singular Hessian foliations, it is clear that they have a strong connection to Loewner’s conjecture in the case n = 2 since (2.4) establishes a method of computing the index of a Loewner vector field where n = 2. We continue with a discussion of the connection between umbilics and singular Hessian foliations. The following lemma appearing in [30] is at the heart of this connection.
Lemma. (Smyth-Xavier, [30]).
(a) Let f : M 2 R3 be a smooth surface. Then on some neighborhood U of each → p M there exist local coordinates z = x+iy with z(p)=o, and a smooth real ∈ function ω(z) such that the Hessian foliations of ω coincide with the principal
foliations of f on U.
(b) Let ω : U R3 be a smooth real function on a neighborhood of o in R2. Then → there exists an immersion f : U R3 of some neighborhood U of p = o in R2 →
22 such that the principal foliations of f coincide with the Hessian foliations of ω on U.
(c) In (a) and (b), the point p is an isolated umbilic of f if and only if o is an
2 isolated zero of the vector field ∂z¯ω, and the index of p as an umbilic of f is
2 equal to half of the index of the vector field ∂z¯ω at o.
Proof. As discussed in the preceding sections of this chapter, the second funda- mental form of the immersion f : M 2 R3 of a 2-manifold into R3 has a pair of → orthogonal principal 1-foliations on the M with the set of umbilics removed. Recall that we have defined N : M 2 S2 to be the unit normal map. Let → Φ: S2 (0, 0, 1) C denote stereographic projection from the point (0, 0, 1). −{ }→ We introduce the Bonnet coordinates in a neighborhood of an umbilical point p. We will begin with the case of an umbilic p with non-zero curvature. One can assume that N(p) = (0, 0, 1). One can choose a neighborhood V of p in M − which is diffeomorphic to U =Φ( N(V )) which contains the origin in C. Thus one has local coordinates z = x + iy for V p. These coordinates are called Bonnet 0 coordinates on M. If λ = 0, one can invert the surface through the unit sphere via the conformal mapΨ . Let fˆ =Ψ f. Then the unit normal map of fˆ is ◦
N, f Nˆ = N 2) *f. − f, f ) *
The second fundamental form of fˆ is given by
IIˆ = f, f II 2 f, N I. ) * − ) *
One can see that the eigenvectors of II are also the eigenvectors of IIˆ . Hence the
23 principal foliations of a neighborhood Uˆ =Φ( Nˆ(Vˆ )) of the origin in C induced by fˆ coincide with those induced by f on U. Furthermore, the eigenvalues of IIˆ p are controlled by f(p). In particular, the eigenvalues of IIˆ are 2 f(p),N(p) . p − ) * Since f(p) is arbitrary, it can be chosen so as to supply non-zero eigenvalues for the second fundamental form at p. We want to show that the principal foliations are in fact a pair of singular Hessian foliations. Consider the Bonnet coordinates defined above. Written in terms of points in the coordinate neighborhood U C, the unit normal map is ⊂ given by (z +¯z, i(z z¯), z 2 1) N(z)= − − | | − . (2.5) 1+ z 2 | | The condition that a vector is principal described in (1.10) is equivalent to
0= f (X) N (X),N (2.6) ) ∗ × ∗ * which can be interpreted as 0 = df dN, N . When expressed in Bonnet coor- ) × * dinates, this condition is equivalent to
0= df dN, N = (f dz + f dz¯) (N dz + N dz¯),N . (2.7) ) × * ) z z¯ × z z¯ *
After some computation, we have
0= f N ,N dz2 + (f N + f N ),N dzdz¯ + f N ,N dz¯2 ) z × z * ) z × z¯ z¯ × z * ) z¯ × z¯ * = f ,N N dz2 +( f ,N N + f ,N N )dzdz¯ + f ,N N dz¯2 ) z z × * ) z z¯ × * ) z¯ z × * ) z¯ z¯ × * (2.8)
If at each point we rotate our frame in R3 so that N(z) = (0, 0, 1), then C −
24 represents the tangent plane at that point. Moreover this implies that since Nz is in the tangent plane, N N = iN and likewise N N = iN . Hence, (2.7) z × z z¯ × − z¯ becomes
0= f ,N N dz2 +( f ,N N + f ,N N )dzdz¯ + f ,N N dz¯2 ) z z × * ) z z¯ × * ) z¯ z × * ) z¯ z¯ × * = f , iN dz2 +( f , iN + f , iN )dzdz¯ + f , iN dz¯2 ) z z* ) z − z¯* ) z¯ z* ) z¯ − z¯* and hence
0= f ,N dz2 +( f ,N f ,N )dzdz¯ f ,N dz¯2. (2.9) ) z z* ) z¯ z*−) z z¯* −) z¯ z¯*
Since f is in the tangent plane, f ,N = 0 in U and one has z ) z *
0=∂ f ,N = f ,N + f ,N z¯) z * ) zz¯ * ) z z¯* = ∂ f ,N = f ,N + f ,N z) z¯ * ) zz¯ * ) z¯ z* meaning f ,N = f ,N = f ,N . ) z z¯* −) zz¯ * ) z¯ z*
Thus (2.9) becomes
0= f ,N dz2 f ,N dz¯2. (2.10) ) z z* −) z¯ z¯*
We now introduce the Bonnet function ω(z) = (1 + zz¯) f, N , recalling the ) * definition of N from (2.5) above. Also recalling that f ,N = f ,N = 0, by ) z * ) z¯ *
25 differentiating one obtains
ω = ∂ ((1 + zz¯) f, N )) z z ) * =¯z f, N + (1 + zz¯)( f ,N + f, N ) ) * ) z * ) z* =¯z f, N + (1 + zz¯) f, N ) * ) z* ω = 2¯z( f ,N + f, N ) + (1 + zz¯)( f ,N + f, N ) zz ) z * ) z* ) z z* ) zz* = f,2¯zN + (1 + zz¯)N + (1 + zz¯) f ,N . (2.11) ) z zz* ) z z*
However if one differentiates the formula in (2.5) for the unit normal map in Bonnet coordinates, one has
(1 + zz¯)(1, i, z¯) z¯(z +¯z, i(z z¯),zz¯ 1) N = − − − − − z (1 + zz¯)2 (1 z¯2, i(1 +z ¯2), 2¯z) = − − (1 + zz¯)2 (1 z¯2, i(1 +z ¯2), 2¯z) N = 2¯z − − zz − (1 + zz¯)3 2¯z = N . (2.12) −1+zz¯ z
Substituting (2.12) into (2.11), one obtains
ω = (1 + zz¯) f ,N . (2.13) zz ) z z*
Similarly, one has
ω = f,2zN + (1 + zz¯)N + (1 + zz¯) f ,N z¯z¯ ) z¯ z¯z¯* ) z¯ z¯* 2z N = N z¯z¯ −(1 + zz¯) z¯
26 and therefore ω = (1 + zz¯) f ,N . (2.14) z¯z¯ ) z¯ z¯*
Combining (2.10), (2.13) and (2.14), one has
2 2 1 2 2 0= f ,N dz f ,N dz¯ = (1 + zz¯)− (ω dz ω dz¯ ) ) z z* −) z¯ z¯* zz − z¯z¯
2 2 and thus Im(ωz¯z¯dz¯ ) = 0. Hence ωz¯z¯ = ρζ . Moreover, ωz¯z¯ = 0 if and only if ω = 0. At such a point p, the condition in (2.6) is satisfied for every X T M, zz ∈ p or equivalently, p is an umbilic. The unit vector field ζ represents either one of the principal foliations of f and, moreover, is a Hessian foliation in either case. Now we proceed to prove the claims of this lemma.
(a) The local coordinates are the Bonnet coordinates described above in a neigh- borhood U of p. The neighborhood U should be chosen so that U p −{ } contains no umbilics, although it is not necessary that p should be an umbilic. As shown above, in Bonnet coordinates the principal foliations of f (or, if λ = 0, fˆ which has identical principal foliations in some neighborhood of p) are in fact the Hessian foliations of ω.
(b) Let U be a neighborhood of o in R2 and let ω : U R be a smooth real-valued → 1 2 function. Letω ˜ :Ψ− (U) S R be defined as ⊂ →
(ω Ψ)(p) ω˜(p)= ◦ . 1+ Ψ(p) 2 | |
1 2 We can considerω ˜ to be defined on an open subsetΨ − (U) of S whereΨ
is stereographic projection. We can also think of S2 as embedded in R3, in
1 which caseΨ − (U) contains the point (0, 0, 1) since U is a neighborhood of −
27 2 3 the origin. Define fc : S R to be →
2 f (p)= S ω˜ + (˜ω(p)+c)ξ(p) c ∇p where S2 is the canonical connection on S2 evaluated at p and defined with ∇p respect to the usual embedding of S2 into R3, ξ is the unit position vector on
2 2 S , and c is a real number. Then for vectors Xp in TpS one has
Xp(fc)=Hω˜ (Xp) + (˜ω + c)Xp,
2 where Hω˜ is the Hessian on S of the functionω ˜. In other words,
df c = Hω˜ + (˜ω + c)I,
and therefore for almost every value of c in R, df c is non-degenerate. Thus one can choose c such that fc is an immersion.
Recalling the definition of the Bonnet function of an immersion, one has that
1 the Bonnet function ωc of fc defined on U (rather than onΨ − (U)) is given by
ω = (1 + zz¯) f ,ξ c ) c * = (1 + zz¯)(˜ω + c)
= ω + c(1 + zz¯),
2 2 2 since ξ is the unit vector normal to S . One notes that ∂z¯ωc = ∂z¯ω, hence the principal foliations of fc (for well-chosen values of c) coincide with the Hessian
28 foliations of ω.
2 (c) As noted following (2.14), one can write ωz¯z¯ = ρζ , where ρ vanishes only at singularities of the Hessian foliations. However, since the Hessian foliations correspond to the principal foliations, ρ(z) = 0 precisely when z corresponds to an umbilic.
This concludes the proof of the lemma.
Our original motivation for the preceding discussion was to show that the Loewner conjecture (for n = 2) implies the Carath´eodory conjecture. In light of the fact that the Local Carath´eodory conjecture is known to imply the Carath´eodory conjecture, the only piece remaining is obtained by the following corollary of the lemma proved above.
Corollary. The Loewner conjecture (n =2) and the the local Carath´eodory con- jecture are equivalent.
Proof. This follows directly from the Lemma above.
29 CHAPTER 3
n A MATRIX REPRESENTATION FOR ∂z¯ f
In this work, we establish a class of index theorems for Loewner vector fields
n of the form ∂z¯ f which extends the work done by Xavier [39] in the case n = 2. Analogous to the proof of the main theorem of [39], our results require an appro-
n priate matrix representation of ∂z¯ f. However, it should be general in the sense that it applies for integers n 2. In fact, the appropriate representation of ∂nf ≥ z¯ 2 is similar to the representation of ∂z¯f in [39] in the sense that it is traceless and symmetric. In the case n = 2, they are identical. However, for n 3 there is ≥ some new structure which presents its own difficulties. Throughout the text, we will assume that n is an integer greater than or equal to 2.
3.1 Some natural identifications
Let ι : C R2 be defined by ι(z) = (Re(z), Im(z)). This map is the inverse of → the canonical identification of R2 and C, namely (x, y) z = x + iy. Since the → conjugate of z is defined and denoted byz ¯ = Re(z) i Im(z), we have formulas − for the real and imaginary parts of z
z +¯z z z¯ x = Re(z)= and y = Im(z)= − . 2 2i
30 These identifications give us a natural identification between partial derivatives in R2 and partial derivatives in C with respect to z andz ¯. The following are the four basic identities:
∂ ∂x ∂ ∂y ∂ 1 ∂ ∂ = + = i (3.1) ∂z ∂z ∂x ∂z ∂y 2 ∂x − ∂y + , ∂ ∂x ∂ ∂y ∂ 1 ∂ ∂ = + = + i (3.2) ∂z¯ ∂z¯ ∂x ∂z¯ ∂y 2 ∂x ∂y + , ∂ ∂z ∂ ∂z¯ ∂ ∂ ∂ = + = + (3.3) ∂x ∂x ∂z ∂x ∂z¯ ∂z ∂z¯ ∂ ∂z ∂ ∂z¯ ∂ 1 ∂ ∂ = + = (3.4) ∂y ∂y ∂z ∂y ∂z¯ i ∂z − ∂z¯ + ,
The differential operator ∂z¯ is called the Cauchy-Riemann operator. It is generalized by the Dolbeault operators ∂ and ∂¯ which act on the (p, q)-forms of a complex manifold M.
Although equations (3.1)-(3.4) are certainly valid for maps of C into itself which are continuously differentiable, there are some consequences specific to real- valued functions, f : R2 = C R. In particular, applying equation (3.2) to some ∼ → ∂f ∂f such f together with ι, and observing that ∂x and ∂y are real-valued, we see that:
∂f 1 ∂f ∂f 1 = + i f. (3.5) ∂z¯ 2 ∂x ∂y 6 2∇ + ,
There are several identifications of C with subspaces of M2(R), the space of 2 2 matrices with real entries. We are interested in two, which will be denoted × by σ, s : C M2(R) and are defined as follows: →
xy σ(x + iy)= , (3.6) y x −
31 and x y − s(x + iy)= . (3.7) yx where x, y R. The former of these (3.6) is used extensively in the development ∈ n of the matrix representation of ∂z¯ f. The latter (3.7) is – generally speaking – the more frequently used of the two. In our context it plays a more limited – but
n still critical – role in the construction of our matrix representation for ∂z¯ f and in proving the main result of this text which appears in Section 4.3. The following matrices will be useful in this chapter and beyond.
10 0 1 − K := σ(1) = and J := s(i)= 0 1 10 − The trace of a matrix A will be denoted by Tr(A). The following proposition develops some straightforward properties of σ and s which will be useful as we continue.
Proposition 3.8. The following hold for σ, s:
(a) σ, s are real-linear as are their inverses.
T (b) A σ(C) M2(R) if and only if A = A and Tr(A) = 0. ∈ ⊂
(c) σ(z)=s(z).K and s(z)=σ(z).K.
(d) σ(z1z2)=σ(z1)Kσ(z2) for all z1,z2 C. ∈
(e) s is a ring isomorphism of C into s(C).
(f ) σ is an additive group (but not ring) isomorphism of C into σ(C).
32 1 1 (g) σ(iz)=J.σ(z) and hence σ− (J.A)=iσ− (A) for all A σ(C). ∈
(h) σ(C) and s(C) are left J-invariant subspaces of M2(R).
Proof. Parts (a), (b) are self-evident. The proofs of the remaining facts are straightforward, but are included for completeness. To obtain (c), we make the computations
x y 10 xy − s(z).K = = = σ(z) yx 0 1 y x − − and σ(z).K =(s(z).K).K = s(z).(K.K)=s(z).
Likewise, for (d)
σ(z1z2)=s(z1z2)K = s(z1)s(z2)K
=(σ(z1)K)(σ(z2)K)K = σ(z1)Kσ(z2)
Regarding (e), letting z1 = x1 + iy1 and z2 = x2 + iy2 we have
x y x y x x y y (x y + x y ) 1 − 1 2 − 2 1 2 − 1 2 − 1 2 2 1 s(z1)s(z2)= = y1 x1 y2 x2 x1y2 + x2y1 x1x2 y1y2 − = s(x x y y + i(x y + x y )) = s(z z ). 1 2 − 1 2 1 2 2 1 1 2
For (f), observe that the real-linearity provided by (a) is enough to show that
σ is an additive group isomorphism. To see that σ is not a ring isomorphism with
33 the usual operations, we note that K = σ(1) = σ(i i), but ·−
01 0 1 − σ(i)= and σ( i)= , 10 − 10 − yet in keeping with (d), σ(i)σ( i)= I which is not equal to K. − − In the case of (g),
0 1 xy yx − − Jσ(z)= = = σ(iz). 10 y x xy − Finally, (h) is a direct result of (g) and (e).
The preceding properties illustrate some of the differences between σ and s which both identify C with M2(R). The action of σ(z) on vectors associated with complex numbers is especially interesting since it requires a conjugation as shown below.
Proposition 3.9. If z1,z2 are in C, then ι(z1z2)=σ(z1)ι(¯z2).
Proof. Let xj = Re(zj) and yj = Im(zj) for j =1, 2. Then
x1x2 y1y2 ι(z z )=ι(x x y y + i(x y + x y )) = − . 1 2 1 2 − 1 2 1 2 2 1 x1y2 + x2y1 However,
x y x x x y y 1 1 2 1 2 − 1 2 σ(z1)ι(¯z2)= = . y1 x1 y2 x1y2 + x2y1 − −
34 3.2 Some useful properties of functions in Cn(D,¯ R)
In this section, we will expose some properties of n-times differentiable func- tions of a complex variable that are real-valued. These functions can be defined on the whole complex plane, but we only require them to be defined in a neigh- borhood of the origin. In fact, after a rescaling of the domain that neighbor- hood can be assumed to contain the closed unit disk D¯, where the unit disk
D = z < 1 C. We will also be interested in the unit circle which will be {| | }⊂ denoted by T = ∂D = z =1 . {| | } In the title of this section, we used the notation Cn(D,¯ R) to refer to these functions. The standard meaning of this entails that functions in this set are n- times continuously differentiable on some open set U containing the closed unit disk. Our goal in presenting these properties is to build up the machinery required to prove the main theorems of Chapter 4. The theorems of that chapter require some additional smoothness near the unit circle. This is not treated directly in the statements of the theorems of this chapter, however the properties that are stated here extend to the additional smoothness specification very easily. The following concise notation is used for partial derivatives:
∂ ∂f ∂ = ; f = ∂ f = (3.10) u ∂u u u ∂u
The following Lemma provides a useful method of modifying the function f which induces a Loewner vector field.
Lemma 3.11. Let U be a neighborhood of D¯ in C = R2. Suppose f Cn(U, R) for ∼ ∈ n n ¯ ˜ n which ∂z¯ f(0) = 0 ∂z¯ f =0on D 0 . Then there exists f C (U, R) satisfying " \{ } ∈
35 the following conditions:
n ˜ n (i) ∂z¯ f = ∂z¯ f
n 1 ˜ (ii) ∂z¯ − f(0) = 0
n 1 (iii) ∂ − f˜ > 0 on D¯ 0 z¯ \{ } ' ' ' ' Proof.' Let α' and β be real numbers. Define
2α n 1 2β n 1 f˜(z)=f(z)+ Re(z − )+ Re(z − z¯). (n 1)! (n 1)! − −
In the following argument, we show that by adjusting α and β, we can produce an f˜ which satisfies conditions (ii) and (iii). Certainly f˜ is in Cn(U, R). A quick n ˜ n computation verifies that ∂z¯ f = ∂z¯ f.
n 1 We will choose α = ∂ − f(0). Then we have − z¯
n 1 n 1 n 1 n 1 ∂ − f˜(0) = ∂ − f(0) + α = ∂ − f(0) ∂ − f(0) = 0. z¯ z¯ z¯ − z¯
It remains only to show (iii). n 1 ˜ Let Bε(0) be a disk of radius ε> 0 in which the Taylor expansion of ∂z¯ − f holds:
n 1 n 1 n 1 n 2 ∂ − f˜(z)=∂ − f˜(0) + z∂ ∂ − f˜(0) +z ¯∂ f˜(0) + O( z ) z¯ z¯ z z¯ z¯ | | n 1 2 = z(∂ ∂ − f(0) + β) +z ¯(0) + O( z ) z z¯ | | =(c + β)z + O( z 2), | |
n 1 where c is defined to be ∂z∂z¯ − f(0).
36 n 1 Clearly as long as β = c is sufficiently large, we have ∂ − f˜(z) = 0 in " − z¯ " B (0) 0 . Now we want to impose positivity on the closed annulus ε \{ }
ε ε A( , 1) = z C : z 1 . 2 { ∈ 2 ≤| |≤ }
n 1 n 1 n 1 Let M be the maximum of ∂ − f(z) ∂ − f(0) . Recalling that α = ∂ − f(0), | z¯ − z¯ | − z¯ one has
n 1 n 1 n 1 ∂ − f˜(z) = ∂ − f(z) ∂ − f(0) + βz z¯ z¯ − z¯ ' ' ' n 1 n 1 ' ' ' ' ∂ − f(z) ∂ − f(0) β' z ' ' ≥ z¯ − z¯ −| | '' ε ' ' 'M' β ' ' (3.12) ≥ − 2 ' ' ' ' ' ' The right hand side (and hence the left hand side) of (3.12) can be made strictly positive by choosing β> 0 to be large enough. Hence an appropriate choice of β
n 1 allows one to say ∂ − f˜(z) > 0 on D¯ 0 . | z¯ | \{ }
In Lemma 3.11, parts (ii) and (iii) can be strengthened to hold for k Cauchy- Riemann derivatives where k =1, . . . , n 1. −
n ¯ n Corollary 3.13. Let f C (U, R) for some neighborhood U of D having ∂z¯ f =0 ∈ " on D¯ 0 . Then there exists f˜ Cn(U, R) satisfying the following: \{ } ∈
n ˜ n (i) ∂z¯ f = ∂z¯ f
(ii) ∂kf˜(0) = 0, where k =1, . . . , n 1 z¯ −
(iii) ∂kf˜ > 0 on D¯ 0 , where k =1, . . . , n 1 z¯ \{ } − ' ' ' ' Proof.' We' argue by induction. Let’s begin by considering the case n = 2. Then k = 1 and this statement is a direct result of Lemma 3.11.
37 The induction hypothesis assumes that that the statement of this corollary is true for n =2,...,N 1. An application of Lemma 3.11 gives us f˜ which satisfies − the hypotheses of this statement for n = N 1. Inductively, we obtain fˆ which − satisfies (ii) and (iii) for n = N. So far, the function fˆ is only guaranteed to be
N 1 2 C − (U, R). However, the method of Lemma 3.11 adds C∞(R , R) functions to f, so the original amount of differentiability is preserved throughout this process.
N Furthermore, none of these operations affect ∂z¯ f.
Remark 3.14. It is pretty clear from the proofs of Lemma 3.11 and Corollary 3.13 that these statements would still hold if one added the condition that such func- tions should be Cn+1 in a neighborhood of the unit circle. Lemma 3.11 modi-
fies f by adding polynomials which are C∞ which retains any finite number of derivatives. Hence the assumption of an additional derivative near the original is consistent with the statements of Lemma 3.11 and Corollary 3.13.
It is also important to note that the f˜retrieved as an “output” of Corollary 3.13 has the same Loewner vector field of order n as the original function f, provided f satisfies the hypotheses of the corollary. Since our ultimate goal is to compute the index of a Loewner vector field for the function f, the invariance of the Loewner vector field with respect to the process described in Corollary 3.13 means we are free to substitute f˜ for f in our index computations.
3.3 The matrix representation
The discussion of the previous sections has introduced all of the necessary
n machinery to construct our matrix representation of ∂z¯ f. It is worth recalling the goals of our matrix representation. In particular, it needs to be traceless and symmetric. The map σ is isomorphic onto its image which
38 consists entirely of symmetric, traceless matrices, so it seems a natural choice to
n apply σ to ∂z¯ f when looking for our desired matrix representation.
Throughout this section, we say that z C A M2(R) if σ(z)=A. How- ∈ 6 ∈ ever, the application of a combinatorial identity produces a non-trivial and very useful formula. Its consequences are the focus of the discussion for the remainder of this chapter. The following lemma implements this strategy.
Lemma 3.15. Suppose f Cn(U, R) for some domain, U R2. Then ∈ ⊂
n 2 − n 1 n n 2 j n 2 j j 2 − ∂ f − J ∂ − − ∂ f (3.16) z¯ 6 j H x y j=0 + , - . /
∆f 0 1 − where (f)=Hf I, and J = s(i)= . H − 2 10 1 Proof. To prove this fact, we will directly apply σ− to the right hand side using several properties of σ and s described in Proposition 3.8. The definition of (f) H from the statement of the lemma, has two immediate consequences, namely
fxx fyy 2fxy 2 (f)= − (3.17) H 2fxy fyy fxx − 1 2σ− ( (f)) = f f +2if H xx − yy xy
Let A denote the right hand side of (3.16). By Proposition 3.8.a, one has
n 2 − 1 1 n 2 j n 2 j j σ− (A)=σ− − J ∂ − − ∂ f j H x y 0 j=0 1 - + , n 2 . / − n 2 1 j n 2 j j = − σ− J ∂ − − ∂ f . (3.18) j H x y j=0 + , - . . //
39 However, if with repeated use of Proposition 3.8.g, one has
1 j n 2 j j j 1 n 2 j j σ− J ∂ − − ∂ f = i σ− ∂ − − ∂ f (3.19) H x y H x y . . // . . // Combining (3.18) and (3.19) and using the definition of ( ), one obtains H ·
n 2 − 1 n 2 j 1 n 2 j j σ− (A)= − i σ− ∂ − − ∂ f j H x y j=0 - + , n 2 . . // − 1 n 2 j n j j n 2 j j+2 n 1 j j+1 = − i ∂ − ∂ ∂ − − ∂ +2i∂ − − ∂ )f 2 j x y − x y x y j=0 + , - . / For the remainder of this proof, we will use the following concise notation:
n k k fk := ∂x− ∂y f. We will also extend the usual definition of the binomial coefficients n k to be equal to 0 if k<0 or k > n. The usual binomial identities still apply, .since/ this is equivalent to extending Pascal’s triangle by inserting zeros in the lattice outside the triangle. This extension is not necessary, but it does simplify the computations. Continuing the computation begun above, we have
n 2 − 1 n 2 j 2σ− (A)= − i (f f +2if ) j j − j+2 j+1 j=0 - + , n 2 − n 2 = − ij f +2if + i2f (3.20) j j j+1 j+2 j=0 + , - . / n n 2 n 2 n 2 = − +2 − + − ijf (3.21) j 2 j 1 j j j=0 - 2+ − , + − , + ,3 n n = ijf (3.22) j j j=0 + , -n n j n j j = i ∂ − ∂ f (3.23) j x y j=0 - + , n n n =(∂x + i∂y) f =2 ∂z¯ f
40 This concludes the proof.
Remark 3.24. Although the above proof is complete, it is useful to highlight a few minor subtleties of the computation. These are listed individually below.
(i) In (3.20), f is simply rewriten as i2f . This accomplishes the task of − j+2 j+2 matching the power of i to the subscript of f.
(ii) In (3.21), the sum was re-indexed to compute a coefficient for fj. Note that
the coefficients for f0 and fn only receive contribution from one of the three binomial coefficients in the sum (since the other two are 0 by our extension
of the binomial coefficient). Likewise f1 and fn 1 only receive contributions − from two of these coefficients.
n n 1 n 1 (iii) In (3.22), the basic binomial identity j = j −1 + −j is applied twice. It − is easy to verify that this identity remains. / true. when/ . the/ binomial coefficients are extended as described in the proof.
n The preceding result gives a matrix representation for ∂z¯ f. It is not yet clear, however, what properties this matrix representation might possess. The following result presents this representation in a more illuminating form. Recall that (g) H is the traceless part of the Hessian of the real-valued function g : C R. → The following statement invokes the object Ker( ) C2(U, R) which is easily H ⊂ computed to be
Ker( )= f C2(U, R):f(x, y)=a(x2 + y2)+bx + cy + d, a, b, c, d R . H { ∈ ∈ }
Theorem 1. Given f Cn(U, R), there exist functions φ and ψ in C2(U, R) ∈ satisfying: ∂nf (φ)+J (ψ) . z¯ 6H H
41 If φ and ψ are chosen to satisfy this condition, then for all φ˜ φ + Ker( ), ∈ H ψ˜ ψ + Ker( ), ∂nf (φ˜)+J (ψ˜). ∈ H z¯ 6H H
Proof. Referring to (3.16), we note that J 2 = I, and hence J 3 = J and − − J 4 = I. Also it is evident that ( ) is linear. We will reorganize the sum in H · (3.16) as follows to obtain expressions for φ and ψ. The even terms contain an alternating factor but no J, while the odd terms contain an alternating factor and every term includes J.
n 2 − n 1 n n 2 j n 2 j j ∂ f 2 − − J ∂ − − ∂ f z¯ 6 j H x y j=0 + , - . / n 1 2 − 1 n 7 8 n 2 j n 2 j j =2− − ( 1) ∂ − − ∂ f 2j − H x y j=0 + , - . / n 2 2 − 1 n 9 : n 2 j n 3 2j 2j+1 +2 − − ( 1) J ∂ − − ∂ f 2j +1 − H x y j=0 + , - . / Continuing, we have
n 1 2 − n 1 n 7 8 n 2 j n 2 j j ∂ f 2 − − ( 1) ∂ − − ∂ f z¯ 6H 2j − x y j=0 + , - n 2 2 − 1 n 9 : n 2 j n 3 2j 2j+1 + J. 2 − − ( 1) ∂ − − ∂ f H 2j +1 − x y j=0 + , - = (φ)+J (ψ) H H
It is clear that φ and ψ so defined are C2(U, R).
Definition 3.25. Given f satisfying the conditions of Theorem 1, one can make a canonical choice of φ and ψ which satisfy the conclusion of the theorem. We
42 denote these canonical choices by φ0 = φ0(f) and ψ0 = ψ0(f), which are defined as follows.
n 1 2 − 1 n 7 8 n 2 j n 2 j j φ =2− − ( 1) ∂ − − ∂ f (3.26) 0 2j − x y j=0 - + , n 2 2 − 1 n 9 : n 2 j n 3 2j 2j+1 ψ =2− − ( 1) ∂ − − ∂ f (3.27) 0 2j +1 − x y j=0 - + ,
Two conventions will be applied to these sums. First, a sum from 0 to 1 (as − in ψ0 in the case n = 2) has no terms and hence is zero. Second, the operators
0 0 ∂x = ∂y = id.
Given f satisfying the hypotheses of Corollary 3.13, we define an operator which sends a function in Cn(D,¯ R) to a C2 function from D¯ into the symmetric, traceless 2 2 matrices with real-valued entries. This operator is denoted byΛ (f) × n and defined as given by Theorem 1:
Definition 3.28. ∂nf Λ (f) := (φ)+J. (ψ), where φ φ (f) + Ker( ) z¯ 6 n H H ∈ 0 H and ψ ψ (f) + Ker( ) per Definition 3.25. ∈ 0 H Let’s take stock of where we are at the moment. We have a matrix representa-
n tion of ∂z¯ f which is a candidate for an analog to the representation used in [39]. More importantly, it performs two subtly non-trivial purposes. First, it relates complex derivatives to real derivatives in a clever way. Second, the real derivatives are in the entries of a matrix which, at points, gives us a well-understood action on vectors in the tangent space.
The following proposition lists some properties of the representation in (3.28) which show that it is indeed the right choice for an analogue for the traceless part of the Hessian as in [39]. Recall that Tr( ) indicates the trace operator. ·
43 Proposition 3.29. Let f be in Cn(U, R) satisfying the conclusions of Corol- lary 3.13. Then the following hold for f:
(i) Λn(f) is symmetric and traceless.
(ii) Λ (f)= (f) (i.e. φ(f)=f and ψ(f) 0). 2 H ≡
(iii) Λ (f)= (f )+J. (f ) (i.e. φ(f)=f and ψ(f)=f ). 3 H x H y x y
Proof. Recall equation (3.17) from Lemma 3.15. This gives us Tr( (φ)) = 0 and H Tr( (ψ)) = 0. Furthermore, from part (h) of Proposition 3.8, it is also true that H Tr(J. (ψ)) = 0. Thus Tr(Λ (f)) = 0. H n T For the same reasons, (Λn(f)) =Λn(f). In the case n = 2, recall from Definition 3.25, that
n 2 2 − 1 n 9 : n 2 j n 3 2j 2j+1 ψ =2− − ( 1) ∂ − − ∂ f. 0 2j +1 − x y j=0 - + ,
However as noted in the definition, when n = 2, one has ψ 0. Thus φ has one 0 ≡ term which is simply f. HenceΛ (f)= (f). 2 H
In the case n = 3, it is clear that φ0 = fx and ψ0 = fy. HenceΛ 3(f)= (f )+J. (f ). H x H y
44 CHAPTER 4
n INDEX FORMULAS FOR ∂z¯ f
In this chapter, we describe a method for proving a class of index theorems relating the index of the Loewner vector field associated with f around an isolated zero and the eigenvectors and eigenvalues of a certain matrix of derivatives of f. These index theorems provide a method of computing the index of such vector fields and casts the Loewner conjecture stated in Chapter 1 in a more geometric light. For clarity, we make the following definition.
Definition 4.1. Let U be a neighborhood of 0 in the complex plane C. A vector
n field ξ : U C is called Loewner of order n N∗ if there exists f in C (U, R) → ∈ such that ξ = ∂nf, ξ(0) = 0 and ξ(z) = 0 for all z = 0. z¯ " "
4.1 Background and Bendixson’s index formula
The methods employed to produce an index formula for Loewner vector fields for n 2 are similar to those used in the case n = 2 as demonstrated in F. Xavier’s ≥ “An Index Formula for Loewner Vector Fields” [39]. We will state the main theorem from this paper, but we need some notation first.
Recall from Chapter 3 that D = z C : z < 1 and T = ∂D. { ∈ | | }
45 Notation 4.2. If L : T M2(R) is a continuous, matrix-valued function on T, → then we define the (possibly empty) set
Σ(L)= p T : L(p) p = λp for some λ R . { ∈ ∈ }
At any p Σ(L), the matrix L(p) has two real eigenvalues λ µ. Hence we can ∈ ≥ partitionΣ( L) into three sets as follows:
Σ (L)= p Σ(L):L(p) p = λp, λ > µ , λ { ∈ } Σ (L)= p Σ(L):L(p) p = µp, µ<λ , and µ { ∈ } Σ (L)= p Σ(L):L(p) p = λp = µp . 0 { ∈ }
Remark 4.3. The definitions of these sets have some useful consequences. As- sume that L is pointwise symmetric and traceless, as is true ofΛ n(f) for example (see Proposition 3.29).
(i) The eigenvalues of L(p) are λ(p), for some λ(p) 0. ± ≥
(ii) p Σ (L) if and only if λ(p)=µ(p) = 0. ∈ 0
(iii) λ = µ = 0 if and only if L(p) = 0.
n (iv) Λn(f)(p) = 0 if and only if ∂z¯ f(p) = 0.
(v) For Loewner vector fields, ∂nf(z) = 0 for z D¯ 0 . z¯ " ∈ \{ }
(vi) From (i)-(v),Σ (Λ (f)) = andΣ(Λ (f)) =Σ (Λ (f)) Σ (Λ (f)). 0 n ∅ n λ n < µ n
The main theorems of this chapter are – after a shift of eigenvalues – extensions of the main result of [39] which is stated below. The original proof of the following
46 result will not be presented. However in Chapter 5 it will be proved as a corollary of the results proved in this chapter.
2 ¯ 3 2 Theorem (Xavier [39]). Let f C (D, R) be C near T with ∂z¯f =0for all ∈ " z =0. Let λ µ denote the eigenvalues of H the Hessian matrix of f, with λ > µ " ≥ f as long as z =0. Assume that the functions ν = λ µ ∂ µ and ν = µ λ ∂ λ " 1 − − r 2 − − r have no zeros on Σλ(Hf ) and Σµ(Hf ) respectively. Then Σ(Hf ) is finite and furthermore
Ind(∂2f,0) = 2 + # p Σ (H ),ν < 0 # p Σ (H ),ν > 0 z¯ { ∈ λ f 1 }− { ∈ λ f 1 } = 2 + # p Σ (H ),ν > 0 # p Σ (H ),ν < 0 . { ∈ µ f 2 }− { ∈ µ f 2 }
Although the original proof of this theorem is not presented in this text, there are several important features of its proof which are duplicated – to greater or lesser extent – in this chapter. It is worthwhile to discuss these similarities, because they explain many of the choices made in the derivation of these index formulas. There are three essential features that we will carry forward in our strategy of proof.
(a) The theorem gives an index formula that lists a “defect term” to Loewner’s conjecture for n = 2. While this defect term arose naturally in the n = 2 case, one must be creative to produce the same effect for all n 2. ≥
(b) The index formula counts radial eigenvectors. The simple geometry of the
circle lends itself to using the index formula to compute examples.
(c) The matrix whose radial eigenvectors are counted should be symmetric and
traceless, since this also makes it relatively easy to compute its eigenvalues and such a matrix has orthogonal eigenvectors.
47 While these characteristics arose naturally in the the n = 2 case, reproducing these features was a substantial challenge both technically and conceptually. Indeed, Chapter 3 was written to replace a single line in [39]. As in [39], we use a well-known theorem of Bendixson to help us calculate the index of a vector field. For the purposes of the Bendixson index theorem below, let C be a smooth, positively oriented, closed curve bounding an open Jordan domain Ω containing a point p. Let ξ be a smooth planar vector field defined on a neighborhood of Ω¯ which does not vanish on Ω¯ p . Assume ξ is tangent to −{ } C at finitely many points.
Definition 4.4. A point q C is said to be ξ-elliptic (resp. ξ-hyperbolic) if ∈ there exists ε> 0 such that the trajectory of ξ passing through q at time zero is contained in Ω( resp. (Ω¯)C) for all times 0 < t <ε. | |
Theorem (Bendixson (1901), [3]). Let eξ and hξ be the number of points in C which are ξ-elliptic and ξ-hyperbolic respectively. Then
e h Ind(ξ, p) = 1 + ξ − ξ (4.5) 2
Proof. See theorem 9.2, ([21], p. 173).
Bendixson’s index theorem can be used to compute the index of the vector
field at the origin in Figure 4.1. The curves γ1,γ2,γ3 and γ4 are trajectories of ξ which are tangent to C at p1,p2,p3 and p4 respectively. The definition above tells us that p1 and p2 are ξ-hyperbolic points while p3 and p4 are ξ-elliptic points. So in the index formula e = h = 2. Hence Ind(ξ, 0) = 1 + (e h )/2 = 1. ξ ξ ξ − ξ Bendixson’s index formula has previously been used in connection with umbil- ics and principal foliations, for example in [39] and [8].
48 Ξ Γ2
p2
Γ1 C p3
Γ3 o p1
Γ4 p4
Figure 4.1. Using Bendixson’s index formula
Remark 4.6. Naively, the Bendixson index theorem yields an index which is an
1 element of 2 Z. For arbitrary, not-necessarily orientable line fields, there is no reason that this index must be an integer. The integer nature of the index comes from considering the way in which the behavior of the vector field changes. In the case of line fields which are integral curves of a Ck vector field (k 1) which ≥ vanishes only at an isolated point (as in the hypotheses of the Bendixson index theorem), the domain is divided into sectors by critical lines which correspond to parabolic trajectories of the vector field. These trajectories are critical when they separate open sets of trajectories of different types. An excellent and thorough account of this is given in [2]. By continuity, the directions of adjacent sectors
49 must agree along the critical curves that bound those sectors. This is sufficient to force the index of the line field (and hence the vector field) to be an integer.
Remark 4.7. Suppose C is a circle centered at the origin and γ :( ε,ε ) R2 − → is a trajectory of ξ tangent to C at a point p. Then p is hyperbolic (resp. elliptic) d2 if and only if γ(t) 2 is negative (resp. positive). This is easy to see in dt2 | | 't=0 ' Figure 4.1. ' ' n We are now ready to formulate two index formulas on ∂z¯ f, using the repre-
n sentation of ∂z¯ f developed in Chapter 3.
4.2 An index formula for n 2 ≥
The first of these two index formulas does not provide as much insight into what is required in order for the conjecture of Loewner to be true. However, if one was interested in computing the index of a specific Loewner vector field, this formula is likely to be simpler, since the operator is.
n n+1 n Theorem 2. Let f C (U, R) be C near T with ∂z¯ f =0for all z =0. ∈ " " Let λ µ denote the eigenvalues of Λ (f), so that λ>µ as long as z =0. Let ≥ n " η =4λ ∂ (∆φ 2λ)+∂ ∆ψ and η =4λ + ∂ (∆φ +2λ)+∂ ∆ψ. Assume that 1 − r − θ 2 r θ
η1,η2 are non-zero on Σ(Λn(f)). Then Σ(Λn(f)) is finite and furthermore
Ind(∂nf,0) = 2 + # p Σ (Λ (f)),η < 0 # p Σ (Λ (f)),η > 0 z¯ { ∈ λ n 1 }− { ∈ λ n 1 } = 2 + # p Σ (Λ (f)),η < 0 # p Σ (Λ (f)),η > 0 . { ∈ µ n 2 }− { ∈ µ n 2 }
The proof of this theorem and the next require two facts summarized below in a lemma. The first of these helps prove the finiteness ofΣ(Λ n(f)). A similar – although less general – argument was given in the proof of the main theorem
50 in Xavier’s article [39]. The second fact establishes a crucial identity about the number of elliptic and hyperbolic λ- and µ- points. We introduce the following useful notations.
Definition 4.8. Suppose X is a Ck vector field for some k 1 which satisfies the ≥ λ µ hypotheses of the Bendixson index formula (Theorem 4.5). Then eX (resp. eX )
λ µ denotes the number of elliptic λ-(resp. µ-) points of X and hX (resp. hX ) denotes the number of hyperbolic λ-(resp. µ-) points of X.
Lemma 4.9. Suppose M =(C∗, , ), where , is the usual inner product on )· ·* )· ·* k C. Suppose that h : C∗ C∗ is C . Let ξ = Jσ(h(z))ι(z) and let k be greater → than the maximum eigenvalue of σ h(p) for all p in T. If ξ p,ξ =0for all p in ◦ ) *" Σ(σ h), then ◦ (i) Σ(σ h) is finite; ◦ e h (ii) ξ − ξ = eλ hλ = eµ hµ. 2 ξ − ξ ξ − ξ The proof of this lemma will be deferred to Section 4.4 after the proof of Theorem 3.
k Proof of Theorem 2. From Corollary 3.13, we can assume that f satisfies ∂z¯ f(0) = 0 and ∂kf > 0 on D¯ 0 , for all k =1, . . . , n 1. z¯ \{ } − Let' g(z')= z 2/2. Thus g(z) = 2∂ g = z, and H = I. Furthermore, ' ' | | ∇ z¯ g 2 n n ∂z¯g = 0 so for all n 2,c R, ∂z¯ (f(z)+cg(z)) = ∂z¯ f(z). For large enough ≥ ∈ c>0, Hf+cg is positive definite. Also for large c, the linear homotopy T (z, s)= (1 s) f(z)+c g(z) does not vanish in D for any 0 s 1 except when z = 0. − ∇ ∇ ≤ ≤ Therefore, replacing f with f + cg (which has no effect on the index calculation
n for ∂z¯ f) one has Ind( f,0) = Ind( g, 0) = 1. (4.10) ∇ ∇
51 These assumptions on f shall be retained throughout the proof. In Chapter 3, the following identifications were established in Theorem 1 and equation (3.5) respectively:Λ (f) := (φ)+J (ψ) ∂nf and f 2∂ f. n H H 6 z¯ ∇ 6 z¯ Recalling Property 3.9, one can make the identification
2(∂nf) ∂ f Λ (f) f, (4.11) z¯ z 6 n ∇ the left-hand side of which is a complex multiplication and the right-hand side of which is a matrix multiplication. In light of (4.10) and (4.11), we change our calculation as follows:
Ind(∂nf,0) = Ind(∂nf ∂ f 2, 0) z¯ z¯ | z¯ | n = Ind(∂z¯ f∂zf,0) + Ind(∂z¯f,0)
= Ind(Λ (f) f,0) + 1. (4.12) n ∇
Let X = J(Λ (f) g), where J = s(i) as in (3.10). Left multiplication by J n ∇ does not affect the index of the vector field, because this action is simply counter- clockwise rotation by π/2 (corresponding to multiplication by i in C as in Propo- sition 3.8 part (g)). This rotation is homotopic to the identity via a homotopy which does not introduce any zeros other than the one at the origin.
From Proposition 3.29,Λ n(f) is known to be traceless and symmetric. The eigenvalues of such a matrix are equal and opposite, henceΛ n(f) is non degenerate
n except when it is 0. However, the identification ofΛ n(f) and ∂z¯ f implies that
Λn(f)(p) is non degenerate for all non-zero p in D. Reprising the arguments
52 leading to (4.10), we obtain
Ind(J(Λ (f) f), 0) = Ind(J(Λ (f) g), 0) = Ind(X, 0). (4.13) n ∇ n ∇
n By (4.12), Ind(∂z¯ f,0) = 1 + Ind(X, 0).
For the remainder of the proof, we will refer to points inΣ λ(Λn(f)) and
Σµ(Λn(f)) as λ- and µ-points respectively. SinceΛ n(f) is a symmetric matrix with distinct eigenvalues, its eigenvectors are orthogonal. Previously it was noted that g(z)=z. Several further observations about g help distill its relationship ∇ ∇ to λ- and µ-points.
(C) g and J g are respectively tangent and normal to the unit circle. ∇ ∇
(R) Since J rotates vectors by π/2, at λ-points,Λ (f)J g = µJ g, and at n ∇ ∇ µ-points,Λ (f)J g = λJ g. n ∇ ∇
(T) X is tangent to T at precisely the points ofΣ(Λ n(f)).
This correspondence allows us to begin using Bendixson’s formula. Indeed (T) implies one needs only to determine whether each point ofΣ(Λ n(f)) is hyperbolic
2 or elliptic. Let p be a point inΣ(Λ n(f)). Let α :( ε,ε ) R be a trajectory − → of X through α(0) = p. Since X is tangent to the unit circle at p, p is a critical point of the function g α. The second derivative of g α at t = 0 indicates ◦ ◦ whether the trajectory reaches a relative minimum or maximum at t = 0. Thus, for small enough ε> 0, if the second derivative is positive (resp. negative) then
α(( ε, 0) (0,ε)) is contained in D¯ C (resp. D) thus p is hyperbolic (resp. elliptic). − ∪ Throughout the computation, we will frequently use the fact that D3g is sym- metric for real-valued, C3 functions g. Specifically, if X,Y and Z are vector fields,
53 D3g(X,Y,Z)=D3g(Y,Z,X)=D3g(Y,X,Z). In other words, the vector fields can be permuted without affecting the computation. Thus we compute the second time-derivative of g α(t) and evaluate at t = 0. ◦
d2 d g(α(t)) = g, JΛ (f) g dt2 dt)∇ n ∇ * d ∆φ = J g, H I g −dt ∇ φ − 2 ∇ 4 + , 5 d ∆ψ g, H I g − dt ∇ ψ − 2 ∇ 4 + , 5 ∆φ = JH X, H I g − g φ − 2 ∇ 4 + , 5 ∆φ J g, H I H X − ∇ φ − 2 g 4 + , 5 ∆φ D3φ(J g, X, g)+ J g, X g − ∇ ∇ ∇ 2 ∇ 4 + , 5 ∆ψ H X, H I g − g ψ − 2 ∇ 4 + , 5 ∆ψ g, H I H X − ∇ ψ − 2 g 4 + , 5 ∆ψ D3ψ( g, X, g)+ g, X g − ∇ ∇ ∇ 2 ∇ 4 + , 5 = H X,JΛ (f) g + g, JΛ (f)H X ) g n ∇ * )∇ n g * ∆ψ D3φ(J g, X, g) D3ψ( g, X, g)+ g, X g − ∇ ∇ − ∇ ∇ ∇ 2 ∇ 4 + , 5
At a λ-point, X = λJ g. At a µ-point, X = µJ g. Also, it was previ- ∇ ∇ ously observed that g(z)=z and Hg = I. So on T, g = J g = 1. Let ∇ |∇ | | ∇ | d2 A(p)= g(α(t)) . Hence at a λ-point p inΣ (Λ (f)), one has dt2 λ λ n 't=0 ' ' ' 1 A(p )=λ(λ µ D3φ(J g, J g, g) D3ψ( g, J g, g)+ ∂ ∆ψ). (4.14) λ − − ∇ ∇ ∇ − ∇ ∇ ∇ 2 θ
54 At a µ-point pµ inΣ µ(Λn(f)), one has
1 A(p )=µ(µ λ D3φ(J g, J g, g) D3ψ( g, J g, g)+ ∂ ∆ψ). (4.15) µ − − ∇ ∇ ∇ − ∇ ∇ ∇ 2 θ
SinceΛ (f) is symmetric and traceless, µ = λ. Thus (4.14) and (4.15) can be n − rewritten as
2A(p ) λ =4λ 2D3φ(J g, J g, g) 2D3ψ( g, J g, g)+∂ ∆ψ (4.16) λ − ∇ ∇ ∇ − ∇ ∇ ∇ θ and
2A(p ) µ =4λ +2D3φ(J g, J g, g)+2D3ψ( g, J g, g) ∂ ∆ψ (4.17) λ ∇ ∇ ∇ ∇ ∇ ∇ − θ respectively. Since λ µ are the eigenvalues ofΛ (f), one has ≥ n
µ J g 2 Λ (f)J g, J g λ J g 2, | ∇ | ≤) n ∇ ∇ *≤ | ∇ | for all p in R2. At λ-points, the inequality on the left is an equality. At µ-points, the inequality on the right is an equality. Hence Λ (f)J g, J g µ J g 2 and ) n ∇ ∇ *− | ∇ | λ J g 2 Λ (f)J g, J g have global minima at λ- and µ- points respectively. | ∇ | −) n ∇ ∇ * Hence, at a λ-point,
0= g( Λ (f)J g, J g µ J g 2) ∇ ) n ∇ ∇ *− | ∇ | ∆φ = D3φ( g, J g, J g)+Dψ3( g, J g, g) g J g 2 ∇ ∇ ∇ ∇ ∇ ∇ −∇ 2 | ∇ | + , + Λ (f)J g, J g + Λ (f)J g, J g g(µ) J g 2 2µ J g 2 ) n ∇ ∇ * ) n ∇ ∇ *−∇ | ∇ | − | ∇ |
55 With some simplification, one has at a λ-point
2(D3φ( g, J g, J g)+Dψ3( g, J g, g)) = ∂ (∆φ 2λ) . (4.18) ∇ ∇ ∇ ∇ ∇ ∇ r −
Similarly, at a µ-point,
0= g(λ J g 2 Λ (f)J g, J g ) ∇ | ∇ | −) n ∇ ∇ * = g(λ) J g 2 +2λ J g 2 D3φ( g, J g, J g) Dψ3( g, J g, g) ∇ | ∇ | | ∇ | − ∇ ∇ ∇ − ∇ ∇ ∇ ∆φ + g J g 2 Λ (f)J g, J g Λ (f)J g, J g ∇ 2 | ∇ | −) n ∇ ∇ *−) n ∇ ∇ * + ,
Thus at a µ-point, one has
2(D3φ( g, J g, J g)+Dψ3( g, J g, g)) = ∂ (∆φ +2λ) . (4.19) ∇ ∇ ∇ ∇ ∇ ∇ r
Recall the definitions of η1 and η2 from the statement of the theorem. Com- bining (4.16) and (4.18), one has
2A(p ) λ =4λ ∂ (∆φ 2λ)+∂ ∆ψ = η (4.20) λ − r − θ 1 while (4.17) and (4.19) combine to give
2A(p ) µ =4λ + ∂ (∆φ +2λ) ∂ ∆ψ = η . (4.21) λ r − θ 2
Recall from Remark 4.7 that the sign of A(pλ)(resp. A(pµ)) determines whether pλ (resp. pµ) is hyperbolic or elliptic. At both λ- and µ-points, λ is positive. Thus
λ-points are hyperbolic when η1 > 0 and such points are elliptic when η1 < 0.
Likewise, µ-points are hyperbolic when η2 > 0 and elliptic when η2 < 0.
56 Having restricted ourselves to the situation in which η1 and η2 are non-zero on Σ(Λ (f)), we pause to note that A(p) = (X g, X ) . From (4.20) and (4.21), n )∇ * |p it is clear that X g, X = 0 onΣ(Λ (f)). Furthermore, the construction of X )∇ *" n is consistent with the conditions on ξ in Lemma 4.9.
Thus Lemma 4.9.(i) implies thatΣ(Λ n(f)) – and hence its subsetsΣ λ(Λn(f)) andΣ µ(Λn(f)) – is a finite set. As a result, one can count the elements of these sets. In particular, eλ = p Σ (Λ (f)) : η < 0 , hλ = p Σ (Λ (f)) : η > 0 , ξ { ∈ λ n 1 } ξ { ∈ λ n 1 } eµ = p Σ (Λ (f)) : η < 0 and hµ = p Σ (Λ (f)) : η > 0 . Combining ξ { ∈ λ n 2 } ξ { ∈ λ n 2 } the results of (4.12) and (4.13), one has
Ind(∂nf,0) = 1 + Ind(J(Λ (f) g), 0) z¯ n ∇ e h = 2 + X − X 2 = 2 + eλ hλ X − X = 2 + eµ hµ X − X
After substituting, one has the formulas in the statement of Theorem 2.
4.3 A defect term for Loewner’s conjecture
At the beginning of Section 4.2, we commented that choosing to use the oper- atorΛ n(f) would make the conditions involved in Theorem 2 simple compared to some other set of conditions which would follow. In Theorem 2, the functions η1 and η2 were defined to detect the ellipticity or hyperbolicity of points inΣ(Λ n(f)). What follows is largely analogous to the arguments of Theorem 2, but additional complexity arises in the functions which detect ellipticity or hyperbolicity due to the choice of a new operator acting on f. This operator is defined below.
57 Definition 4.22. Recalling Definition 3.28 and (3.7), one makes the following definition at points z in C:
n 2 (f)(z) :=Λ (f)s(z − ). Ln n
Notice that when n = 2, s(z0)=s(1) = I and therefore (f)=Λ (f). Ln n Recalling the definition ofΛ (f), one finds that (f)=Λ (f)= (f), the n Ln n H traceless part of the Hessian as in [39]. The goal of introducing this new operator is to establish a defect term for the Loewner conjecture. Theorem 2 is useful, because it provides a method of directly computing the index of a Loewner vector field, but it does not shed much light on what would be necessary for the Loewner conjecture to be true. Now we introduce a second index formula, our main result, which counts eigen- vectors of the operator (f). The proof of this theorem is similar in outline to Ln the proof of Theorem 2, but full details are provided for completeness.
n n+1 n Theorem 3. Let f C (U, R) be C near T with ∂z¯ f =0for all z =0. ∈ " " Let λ µ denote the eigenvalues of (f), so that λ > µ as long as z =0. ≥ Ln " n 2 n 2 Let ζ =4λ 2∂ λ Im(z − )(∂ ∆φ + ∂ ∆ψ) Re(z − )(∂ ∆φ ∂ ∆ψ) and 1 − r − θ r − r − θ n 2 n 2 ζ =4λ +2∂ λ + Im(z − )(∂ ∆ψ + ∂ ∆φ) + Re(z − )(∂ ∆φ ∂ ∆ψ). Assume 2 r r θ r − θ that ζ ,ζ are non-zero on Σ( (f)). Then Σ( (f)) is finite and furthermore 1 2 Ln Ln
Ind(∂nf,0) = n +# p Σ ( (f)),ζ < 0 # p Σ ( (f)),ζ > 0 z¯ { ∈ λ Ln 1 }− { ∈ λ Ln 1 } = n +# p Σ ( (f)),ζ < 0 # p Σ ( (f)),ζ > 0 . { ∈ µ Ln 2 }− { ∈ µ Ln 2 }
Proof of Theorem 3. Let g(z)= z 2/2. As in Theorem 2, we can assume that f | | satisfies the following:
58 ∂kf(0) = 0 and ∂kf > 0 on D¯ 0 , for all k =1, . . . , n 1; • z¯ z¯ \{ } − ' ' H is positive definite;' ' • f
Ind( f,0) = Ind( g, 0) = 1. • ∇ ∇
As in (4.12), it is beneficial to employ the algebra of index computation. How- ever, we want to manufacture an integer term to replace the 1 in the earlier remark with an n 1. Thus we need to produce a vector field whose index is n 1. One − − n 1 obvious example of such a vector field is z − . Using this example as inspiration, we make the following computation:
n n 2n 2 Ind(∂ f,0) = Ind(∂ f ∂ f − , 0) z¯ z¯ | z¯ | n n 1 n 1 = Ind(∂z¯ f(∂zf) − , 0) + Ind((∂z¯f) − , 0)
n 1 = Ind(Λ (f)ι((∂ f) − ), 0) + n 1 n z¯ − n 2 = Ind(Λ (f)s((∂ f) − )ι(∂ f), 0) + n 1 n z¯ z¯ − n 2 = Ind(Λ (f)s((∂ f) − )ι(∂ g), 0) + n 1 n z¯ z¯ − = Ind( (f) g, 0) + n 1, (4.23) Ln ∇ −
Now, as in the previous theorem, we have a candidate for a natural vector field whose index can be computed. In this case, we let X = J( (f) g), where Ln ∇ J = s(1) as in (3.10). Left multiplication by J does not affect the index of the vector field, because this action is simply counterclockwise rotation by π/2
(corresponding to multiplication by i in C as in Proposition 3.8 part (g)). This rotation is homotopic to the identity via a homotopy which does not introduce any zeros other than the one at the origin. LikeΛ (f), (f) is easily observed to be traceless and symmetric. The eigen- n Ln
59 values of such a matrix are of equal magnitude and opposite sign, hence (f) is Ln non-degenerate unless it is 0. As withΛ (f), (f) is only 0 when z = 0. Thus n Ln
Ind(∂nf,0) = n 1 + Ind(X, 0). (4.24) z¯ −
For the remainder of the proof, we will refer to points inΣ ( (f)) and λ Ln Σ ( (f)) as λ- and µ-points respectively. Since (f) is a symmetric matrix µ Ln Ln with distinct eigenvalues, its eigenvectors are orthogonal. Consequently, we can reprise observations (C), (R) and (T) as follows:
(C’) g and J g are respectively normal and tangent to the unit circle. ∇ ∇
(R’) Since J rotates vectors by π/2, at λ-points, (f)J g = µJ g, and at Ln ∇ ∇ µ-points, (f)J g = λJ g. Ln ∇ ∇
(T’) X is tangent to T at precisely the points ofΣ( n(f)). L
This correspondence allows us to begin using Bendixson’s formula. Indeed (T’) implies that one only needs to determine whether each point ofΣ( (f)) is Ln 2 hyperbolic or elliptic. Let p be inΣ( n(f)). Let α :( ε,ε ) R be a trajectory L − → of X through α(0) = p. Since X is tangent to the unit circle at p, p is a critical point of the function g α. The second derivative of g α at t = 0 indicates ◦ ◦ whether the trajectory reaches a relative minimum or maximum at t = 0. Thus, for small enough ε> 0, if the second derivative is positive (resp. negative) then
α(( ε, 0) (0,ε)) is contained in D¯ C (resp. D) thus p is hyperbolic (resp. elliptic). − ∪ So we need to compute the second time-derivative of g α(t) and evaluate ◦ it at t = 0. This computation is somewhat different than the corresponding
n 2 computation in Theorem 2 due to the s(z − ) term in the definition of (f). It Ln
60 begins similarly, by noting
d2 g(α(t)) = X g, X = g, X + g, X . (4.25) dt2 )∇ * )∇X ∇ * )∇ ∇X *
n 2 f Now let Y = s(z − ) g (i.e. X = JΛ (f)Y ). Recalling that (f)=H ∇ I, ∇ n H f − 2 one has
d2 g(α(t)) = H X,X + g, (JΛ (f)Y ) dt2 ) g * )∇ ∇X n * = X,X + g, ((J (φ) (ψ)) Y ) ) * )∇ ∇X H −H *
Noting that J ( )= ( ) J, we continue with H · −H ·
d2 g(α(t)) = X,X g, ( (φ) JY) g, ( (ψ) Y ) dt2 ) * − )∇ ∇X H * − )∇ ∇X H * = X,X g, (φ) (JY) g, (ψ) Y ) * − )∇ H ∇X * − )∇ H ∇X * D3φ( g, X, JY ) D3ψ( g, X, Y ) − ∇ − ∇ ∆φ ∆ψ + X g, JY + X g, Y 2 )∇ * 2 )∇ * + , + , = X,X + g, JΛ (f) Y ) * )∇ n ∇X * D3φ( g, X, JY ) D3ψ( g, X, Y ) − ∇ − ∇ ∆φ ∆ψ + X g, JY + X g, Y (4.26) 2 )∇ * 2 )∇ * + , + ,
At a λ-point, X = λJ g, while at a µ-point, X = µJ g. Once again, we will ∇ ∇ d2 define A(p)= g(α(t)) . We will carefully substitute these expressions into dt2 't=0 ' (4.26). There is one small' but important computation to make prior to carrying '
61 this substitution through to its consequences.
n 1 n 1 J gY =( gy∂x + gx∂y)Y =( gy∂x + gx∂y)(Re(z − )∂x + Im(z − )∂y) ∇ ∇ − − n 2 n 2 n 2 n 2 =(n 1)(( g Re(z − ) g Im(z − ))∂ +( g Im(z − )+g Re(z − ))∂ − − y − x x − y x y n 2 =(n 1)Js(z − ) g =(n 1)JY (4.27) − ∇ −
Let’s start with a λ- point p inΣ ( (f)). λ λ Ln
A(pλ)= λJ g,λJ g + g, JΛn(f) λJ gY ) ∇ ∇ * )∇ ∇ ∇ * D3φ( g,λJ g, JY ) D3ψ( g,λJ g, Y ) − ∇ ∇ − ∇ ∇ ∆φ ∆ψ + λ(J g) g, JY + λ(J g) g, Y ∇ 2 )∇ * ∇ 2 )∇ * + , + , 2 = λ(λ J g + g, JΛn(f) J gY | ∇ | )∇ ∇ ∇ * D3φ( g, J g, JY ) D3ψ( g, J g, Y ) − ∇ ∇ − ∇ ∇ ∆φ ∆ψ +(J g) g, JY +(J g) g, Y ) (4.28) ∇ 2 )∇ * ∇ 2 )∇ * + , + ,
Applying (4.27), the fact that if A is a symmetric traceless 2 2 matrix, then × AJ = JA and the fact that g is a λ-eigenvector for (f), one has − ∇ Ln
g, JΛn(f) J gY = g, (n 1)JΛn(f)JY )∇ ∇ ∇ * )∇ − * =(n 1) g, JJΛ (f)Y − )∇ − n * =(n 1) g, Λ (f)Y − )∇ n * =(n 1) g, (f) g − )∇ Ln ∇ * =(n 1) g,λ g =(n 1)λ. (4.29) − )∇ ∇ * −
62 Combining (4.28) and (4.29), one has
A(p )=λ(nλ D3φ( g, J g, JY ) D3ψ( g, J g, Y ) λ − ∇ ∇ − ∇ ∇ ∆φ ∆ψ +(J g) g, JY +(J g) g, Y ) ∇ 2 )∇ * ∇ 2 )∇ * + , + , = λ(nλ D3φ( g, J g, JY ) D3ψ( g, J g, Y ) − ∇ ∇ − ∇ ∇ 1 + ( g, JY ∂ ∆φ + g, Y ∂ ∆ψ)), (4.30) 2 )∇ * θ )∇ * θ since J g corresponds to the tangential derivative ∂ . ∇ θ
Similarly, at a µ-point pµ inΣ µ(Λn(f)), one has
A(p )=µ(nµ D3φ( g, J g, JY ) D3ψ( g, J g, Y ) µ − ∇ ∇ − ∇ ∇ 1 + ( g, JY ∂ ∆φ + g, Y ∂ ∆ψ)) (4.31) 2 )∇ * θ )∇ * θ
Since λ and µ are the eigenvalues of (f), Ln
µ J g 2 (f)J g, J g λ J g 2, | ∇ | ≤ )Ln ∇ ∇ *≤ | ∇ | for all p in the domain of f. At λ-(resp. µ-) points, the inequality on the left (resp. right) becomes equality. Hence, the functions (f)J g, J g µ J g 2 )Ln ∇ ∇ *− | ∇ | and λ J g 2 (f)J g, J g have global minima at λ- and µ- points respec- | ∇ | − )Ln ∇ ∇ *
63 tively. We can now make the following computation at a λ-point:
0= g( (f)J g, J g µ J g 2) ∇ )Ln ∇ ∇ *− | ∇ | 2 = g(Λn(f)JY),J g + n(f)J g, g(J g) ( g)(µ J g ) )∇∇ ∇ * )L ∇ ∇∇ ∇ *− ∇ | ∇ |
= ( g(Λn(f)))JY,J g + Λn(f)( g(JY)),J g ) ∇∇ ∇ * ) ∇∇ ∇ * + (f)J g, J g ( g)(µ) J g 2 2µ J g 2 )Ln ∇ ∇ *− ∇ | ∇ | − | ∇ |
= ( g (φ))JY,J g + ( g (ψ))JY,J g ) ∇∇ H ∇ * ) ∇∇ H ∇ * +(n 1) Λ (f)JY,J g + Λ (f)JY,J g 2µ ∂ µ − ) n ∇ * ) n ∇ *− − r = D3φ( g, JY, J g)+D3ψ( g, Y, J g) ∇ ∇ ∇ ∇ ∆φ ∆ψ g JY,J g g Y,J g (n 2)λ + ∂ λ −∇ 2 ) ∇ *−∇ 2 ) ∇ *− − r + , + ,
Therefore, at a λ-point
1 D3φ( g, J g, JY )+D3ψ( g, J g, Y )= ( Y, g ∂ ∆φ + Y,J g ∂ ∆ψ) ∇ ∇ ∇ ∇ 2 ) ∇ * r ) ∇ * r ∂ λ +(n 2)λ. (4.32) − r −
After performing similar computations at a µ-point, one has
1 D3φ( g, J g, JY )+D3ψ( g, J g, Y )= ( Y, g ∂ ∆φ + Y,J g ∂ ∆ψ) ∇ ∇ ∇ ∇ 2 ) ∇ * r ) ∇ * r + ∂ λ (n 2)λ. (4.33) r − −
Recall the definitions of ζ1 and ζ2 from the statement of the theorem. Also
2 n 2 2 n 2 note that g, Y = z Re(z − ) and g, JY = z Im(z − ). Combining )∇ * | | )∇ * −| |
64 (4.30) and (4.32), one has
2A(pλ) n 2 n 2 =4λ 2∂ λ Im(z − )(∂ ∆φ + ∂ ∆ψ) Re(z − )(∂ ∆φ ∂ ∆ψ)=ζ λ − r − θ r − r − θ 1 (4.34) while (4.31) and (4.33) combine to give
2A(pλ) n 2 n 2 =4λ 2∂ λ + Im(z − )(∂ ∆φ + ∂ ∆ψ) + Re(z − )(∂ ∆φ ∂ ∆ψ)=ζ . λ − r θ r r − θ 2 (4.35)
Recall from Remark 4.7 that the sign of the second derivative A(pλ) or A(pµ) determines whether its respective point is hyperbolic or elliptic. At both λ- and
µ-points, λ is positive. Thus λ-points are hyperbolic when ζ1 > 0 and such points are elliptic when ζ1 < 0. Likewise, µ-points are hyperbolic when ζ2 > 0 and elliptic when ζ2 < 0. In the hypotheses of the theorem, we restricted ourselves to the situation in which ζ and ζ are non-zero onΣ( (f)). We pause to note that A(p)= 1 2 Ln (X g, X ) . From (4.34) and (4.35), it is clear that X g, X = 0 onΣ( (f)). )∇ * |p )∇ *" Ln Furthermore, the construction of X is consistent with the conditions on ξ in Lemma 4.9. Thus Lemma 4.9.(i) implies thatΣ( (f)) – and hence its subsetsΣ ( (f)) Ln λ Ln andΣ ( (f)) – is a finite set. It follows that one can count the elements of these µ Ln sets. In particular,
eλ = p Σ ( (f)) : ζ < 0 ,hλ = p Σ ( (f)) : ζ > 0 , ξ { ∈ λ Ln 1 } ξ { ∈ λ Ln 1 } eµ = p Σ ( (f)) : ζ < 0 and hµ = p Σ ( (f)) : ζ > 0 . ξ { ∈ λ Ln 2 } ξ { ∈ λ Ln 2 }
65 Recalling (4.24), one has
Ind(∂nf,0) = n 1 + Ind(J( (f) g), 0) z¯ − Ln ∇ e h = n + X − X 2 = n + eλ hλ X − X = n + eµ hµ X − X
After substituting, one has the result.
4.4 Proof of Lemma 4.9
Lemma 4.9 was stated somewhat below the statement of Theorem 2. Its proof, which was deferred above, is now given.
Proof of Lemma 4.9. The vector field ξ is defined in terms of z C. This is a ∈ minor abuse of notation, in the sense that it is assumed that z is identified in the usual way with a point in R2. Also, it is worth noting that g(z) is identified ∇ with z in the sense of vector identification as well.
(i) Define the map F : R2 R2 by F (q) = ( q 2, ξ(q),q ). Note that q is in → | | ) * Σ(σ h) if and only if F (q) = (1, ξ(q),q ) = (1,a(q) Jq, q ) = (1, 0). But F is ◦ ) * ) * 1 continuous, soΣ( σ h)=F − ( (1, 0) ) is closed. ◦ { }
SupposeΣ( σ h) is infinite. Then there exists some infinite sequence pn n 1 ◦ { } ≥ of distinct elements ofΣ( σ h). Since this sequence is a subset of the compact ◦ set T, passing to a subsequence one has that pnk p. This p is inΣ( σ h) → ◦ sinceΣ( σ h) is closed. To contradict the infiniteness ofΣ( σ h), it is therefore ◦ ◦ sufficient to prove that F is a local diffeomorphism at p. For this purpose, we compute the Jacobian of F at p. We will use the basis
66 e1 = Jξ(p), e2 = ξ(p). To begin with, we compute
2 2 2 i b = (e ) q = q ,ξ(q) = 2p, a(p)J − p =2a(p)δ . (4.36) 1i i | | q=p )∇| | * q=p ) * 1i . /' ' ' ' Since b = 0, det DF(p) = 2a(p)b . However, b = ξ(p) q,ξ , which was 12 22 22 ) *|q=p assumed to be non-zero in the hypotheses. The inverse function theorem implies that F is a local diffeomorphism at p and henceΣ( σ h) is finite, thus proving ◦ part (i).
(ii) Let λ(p) to be the non-negative eigenvalue of σ h at p. This is well-defined ◦ since σ h is traceless and symmetric. ◦ Let λmax = max λ(p):p T . Fix k >λ max. Define another vector field on { ∈ } C as follows:Ξ( p) := ξ(p)+kJp = J(σ h + kI)p. The eigenvalues of σ h + kI ◦ ◦ are k + λ and k λ. Both eigenvalues are positive, hence σ h + kI is positive − ◦ definite. The positive definiteness of σ h + kI means that there exists a non-vanishing ◦ homotopy from Ξ(via JΞ) to the radial vector field p. Since the index of a − vector field at a point is invariant under such homotopies, one has Ind(Ξ, 0) = 1.
λ µ Bendixson’s index formula implies that eΞ = hΞ. However, eΞ = eΞ + eΞ and
λ µ hΞ = hΞ + hΞ. Therefore, eλ hλ = hµ eµ . (4.37) Ξ − Ξ Ξ − Ξ
In the proof of Theorems 2 and 3, we computed the second derivative of tra- jectories of a vector field through a point on T to determine whether that point
67 was hyperbolic or elliptic. Recalling (4.25), one has
A (p) =Ξ p, Ξ =Ξ p,ξ + kJp =Ξ p,ξ +Ξ( k p, Jp ) =Ξ p,ξ Ξ ) *|p ) *|p ) *|p ) * |p ) *|p (4.38) since p, Jp = 0. ) * At a point p inΣ( σ h), letting α = λ represent the eigenvalue associated ◦ ± with p, one has
k + α A (p)=(k + α)(Jp) p,ξ = A (p). (4.39) Ξ ) * α ξ
Hence at λ-points, AΞ(p) and Aξ(p) have the same sign; at µ-points, they have
λ λ λ λ opposite signs. Therefore, statement (4.39) implies that eΞ = eξ and hΞ = hξ ; µ µ µ µ meanwhile eΞ = hξ and hΞ = eξ . Combining this with (4.37) yields the result in Lemma 4.9.ii.
68 CHAPTER 5
FURTHER REMARKS
In this chapter, we present some consequences of the index formulas from Chapter 4. First, we formally state the connection between Theorem 4.3 and the Loewner conjecture. We also prove the main theorem of [39] as a corollary of our results and state the connection between this theorem and the Carath´eodory conjecture. Finally, we make some comments about the index of certain Toeplitz operators.
5.1 Application to the Loewner conjecture
In Chapter 1, we introduced a conjecture of Loewner. We restate it below:
n ¯ n Conjecture (Loewner (c. 1950)). If f C (D, R) and ∂z¯ f =0for z =0, then ∈ " " n the index at zero of the vector field given in complex form by ∂z¯ f is at most n.
Theorem 3 in Chapter 4 provides a necessary and sufficient analytic condition for the Loewner conjecture to be true. It is stated in the following corollary.
Corollary 5.1. The Loewner conjecture holds if and only if for every f satisfying the hypotheses in the statement of Theorem 3
# p Σ ( (f)),ζ < 0 # p Σ ( (f)),ζ > 0 , { ∈ λ Ln 1 }≤ { ∈ λ Ln 1 }
69 n 2 n 2 where ζ =4λ 2∂ λ Im(z − )(∂ ∆φ + ∂ ∆ψ) Re(z − )(∂ ∆φ ∂ ∆ψ). 1 − r − θ r − r − θ
5.2 Xavier’s index theorem
When n = 2, the two main theorems of Chapter 4 are identical. This is mostly the result of the fact noted in Proposition 3.29.ii that for n = 2, ψ 0. Either ≡ of these theorems and Lemma 4.9.(ii) imply the theorem of Xavier that was the inspiration for this work.
2 ¯ 3 2 Theorem (Xavier [39]). Let f C (D, R) be C near T with ∂z¯f =0for all ∈ " z =0. Let λ µ denote the eigenvalues of H the Hessian matrix of f, with " ≥ f λ > µ as long as z =0. Assume that the functions ν = λ µ ∂ µ and ν = " 1 − − r 2 µ λ ∂ λ have no zeros on Σ (H ) and Σ (H ) respectively. Then Σ is finite − − r λ f µ f and furthermore
Ind(∂2f,0) = 2 + # p Σ (H ),ν < 0 # p Σ (H ),ν > 0 z¯ { ∈ λ f 1 }− { ∈ λ f 1 } = 2 + # p Σ (H ),ν > 0 # p Σ (H ),ν < 0 . { ∈ µ f 2 }− { ∈ µ f 2 }
Proof. We will prove this result as a corollary of Theorems 2 and 3. Proposition 3.29 implies that φ = f and ψ = 0. Add to this the result of Theorem 2 (or 3). After some simplification, one has
Ind(∂2f,0) = 2 + # p Σ (Λ (f)) : 4λ +2∂ λ ∂ ∆f<0 z¯ { ∈ λ 2 r − r } # p Σ (Λ (f)) : 4λ +2∂ λ ∂ ∆f>0 − { ∈ λ 2 r − r } = 2 + # p Σ (Λ (f)) : 4λ +2∂ λ + ∂ ∆f<0 { ∈ µ 2 r r } # p Σ (Λ (f)) : 4λ +2∂ λ + ∂ ∆f>0 . − { ∈ µ 2 r r }
70 We need to restate the above in terms of Hf =Λ2(f) + (∆f/2)I. At any point p, the eigenvectors of Hf are the same as the eigenvectors ofΛ 2(f). The eigenvalues of Hf are λ" = λ+(∆f)/2 and µ" = µ+(∆f)/2. In particular, λ" >µ" and thusΣ λ(Hf ) =Σ λ(Λ2(f)) andΣ µ(Hf ) =Σ µ(Λ2(f)). Also we recall that
µ = λ. Thus, 4λ +2∂ λ ∂ ∆f =2λ 2µ ∂ µ" = 2(λ" µ" ∂ µ") and − r − r − − r − − r 4λ +2∂ λ + ∂ ∆f =2λ 2µ + ∂ λ" = 2(µ" λ" ∂ λ"). r r − r − − − r
5.3 Application to the Carath´eodory conjecture
Our index theorems also bear on the longstanding Carath´eodory conjecture about the minimum number of umbilical points on a smooth, convex, embedded 2-sphere. The following does not strengthen the conclusions of [39] in this regard. However since the Carath´eodory conjecture is part of the motivation for studying Loewner’s conjecture, it seems worthwhile to recall those conclusions.
Conjecture (Carath´eodory). Every convex C2 embedding of S2 into R3 has at least two umbilic points.
Corollary 5.2 (Xavier [39]). The Carath´eodory conjecture holds if and only if for every f satisfying the hypotheses on f in Xavier’s index theorem
# p Σ (H ),ν < 0 # p Σ (H ), ν> 0 , { ∈ λ f 1 }≤ { ∈ λ f } where ν = λ µ ∂ µ. − − r This statement is somewhat different from what one would obtain directly from Theorem 3. The difference is that in the statement of the corollary, Hf is the function that maps T to M2(R) rather than (f). However, as noted in H previous chapters, this difference only shifts eigenvalues. It does not change the
71 eigenvectors or their number. It is also preferable to think in terms of Hf , since it is a more natural object.
5.4 Application to Toeplitz operators
In the following, L2(T) denotes the space of functions f : T C which → are square integrable and H2 represents the Hardy space of square integrable functions, defined by
2π H2 = f L2(T): f(eiθ)einθdθ = 0 for all n>0 . ∈ 6 (0 7
Let P be the projection map from L2(T) onto its (closed) subspace H2.
2 Definition 5.3. For φ in L∞(T), the Toeplitz operator Tφ on H is defined by
Tφf = P (φf)
2 2 2 for f in H . Hence, Tφ sends functions in H to functions in H .
Toeplitz operators have a rich theory in part owing to the fact that – under the correct circumstances – they are Fredholm operators. In this discussion, H , L(H ) and LC(H ) denote a Hilbert space, the bounded linear operators on H and the compact operators on H respectively.
The subspace LC(H ) of compact linear operators on a Hilbert space has some interesting and useful properties. If H is finite dimensional, LC(H )=L(H ).
Otherwise, LC(H ) is a minimal two-sided ideal of L(H ). This allows us to discuss the quotient L(H )/LC(H ) and the natural projection map
L( ) π : L(H ) H ! . → LC(H )
72 Recall the definitions of Fredholm operators and their indices.
Definition 5.4 (Fredholm operator, p.113, [7]). If H is a Hilbert space, then T in L(H ) is a Fredholm operator if π(T ) is an invertible element of L(H )/LC(H ). The collection of Fredholm operators on H is denoted by F(H ).
It is worth noting that if T is a Fredholm operator on the Hilbert space H , then both dim ker T and dim ker T ∗ are finite. The argument for this goes as follows. If T is Fredholm, then π(T ) is an invertible element of L(H )/LC(H ). Hence there exists A L(H ) such that AT = I mod LC(H ). Thus, AT = I+K ∈ where K is a compact linear operator on H . Suppose that f is in ker T . Then (I + K)f = 0 as well, and furthermore Kf = f. To summarize, −
ker T ker AT = ker(I + T ) Range K. ⊂ ⊂
However since K is compact, the range of K cannot contain a closed infinite- dimensional subspace. The argument is similar for T ∗.
Definition 5.5 (Index of a Fredholm operator, p.115, [7]). If H is a Hilbert space, the classical index j is the the function defined from F(H ) to Z such that j(T ) = dim ker T dim ker T ∗. −
The following theorem gives some sufficient conditions for Tφ to be a Fredholm operator and how its index can be computed.
Theorem (Theorem 7.26, p.165, [7]). If φ is a continuous function on T, then the operator Tφ is a Fredholm operator if and only if φ does not vanish and in this case j(Tφ), the classical index of the Fredholm operator Tφ, is equal to minus the winding number with respect to the origin of the curve traced out by φ.
73 The winding number of the curve traced out by φ is precisely the index of the vector field represented by φ. In particular, Theorem 4.3 can be restated as follows.
n n+1 n Corollary 5.6. Let f C (U, R) be C near T with ∂z¯ f =0for all z =0. ∈ " " Let λ µ denote the eigenvalues of (f), so that λ>µ as long as z =0. Let ≥ Ln " n 2 n 2 ζ =4λ 2∂ λ Im(z − )(∂ ∆φ + ∂ ∆ψ) Re(z − )(∂ ∆φ ∂ ∆ψ). Assume that − r − θ r − r − θ ζ is non-zero on Σ( (f)). Then Σ( (f)) is finite and Ln Ln
j(T n )=n +# p Σ ( (f)), ζ< 0 # p Σ ( (f)), ζ> 0 . − ∂z¯ f { ∈ λ Ln }− { ∈ λ Ln }
This formulation of a defect term for Loewner’s conjecture which employs Toeplitz operators is intriguing in the sense that the Cauchy-Riemann operator plays two prominent roles. The holomorphicity that surrounds the notion of the Toeplitz operator is balanced against the obvious role of the Cauchy-Riemann op- erator in the statement of Loewner’s conjecture. One hopes that this relationship could somehow be exploited to make further progress on this question.
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