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AN INDEX FORMULA FOR LOEWNER VECTOR FIELDS

Frederico Xavier

Abstract. Let f be C 2 real-valued function de?ned on a neighborhood of 0 in R2 , ? f = 0 for ?z 2 z = 0. A conjecture attributed to Loewner, related to umbilical points on surfaces, states that the index at 0 of the vector ?eld given in complex notation by establish a formula that computes the index of

?2f ?z 2 ?2f ?z 2

2

is at most two. In this paper we

from data about the hessian of f .

§1 Introduction. The well-known Carath?odory conjecture in classical di?erential geometry states that any e smoothly immersed sphere in R3 must have at least two umbilical points. Several solutions to this problem have been given over the years, all in the case of analytic immersions [1], [3], [4], [6], [7], [11] (for related work on umbilics, see [2], [8], [9], [10]). In these papers the aim was to prove a stronger form of the Carath?odory conjecture namely, that the local index of the ?elds of lines e of curvature near an isolated umbilic cannot exceed one. Indeed, since the Euler characteristic of S 2 is two, the Poincar?-Hopf index theorem implies that in an immersed sphere with a single e umbilical point the ?elds of principal directions would have index two, thus contradicting the conclusion in the local conjecture. The path followed in [1], [3], [4], [6], [7], [11] was through a conjecture attributed to Loewner, involving functions in the plane and the square of the Cauchy-Riemann operator. Let B be the open unit ball in R2 centered at 0, T = ?B. The Loewner Conjecture. If f ∈ C 2 (B, R) and the vector ?eld given in complex notation by

?2f ?z 2 ?2f ?z 2

= 0 for z = 0, then the index at zero of

is at most two.

The non-vanishing condition in the Loewner conjecture implies that the hessian matrix Hf of f is not a multiple of the identity for z = 0. At a point other than z = 0, Hf has precisely two diagonalizing directions. Thus, one has two orthogonal one-dimensional foliations, possibly with a singularity at 0. The geometric content of the conjecture is that the index at 0 of either 2 of these ?elds of eigendirections of Hf is at most one (in fact, the index of ? f is always twice ?z 2 the index of the ?elds of eigendirections of Hf [10]).

Typeset by AMS-TEX

2

Frederico Xavier

The C 2 Loewner conjecture remains daunting, especially if one takes into account the complexity of the proofs in the analytic case that have been o?ered in the last seventy years or so. Nevertheless - and this is the aim of the present paper -, it is possible to establish a fairly explicit 2 formula that computes the index of ? f from data about the hessian of f : ?z 2 Theorem 1. Let Let f ∈ C 2 (B, R) be C 3 near T , ? f = 0 for z = 0. Denote by λ, ?, λ ≥ ?, ?z 2 the eigenvalues of Hf , so that λ > ? for z = 0. Let Σ = Σλ ∪ Σ? , where Σλ (resp., Σ? ) is the set of points p ∈ T for which Hf (p)p = λ(p)p (resp., Hf (p)p = ?(p)p). Assume that the function λ ? ? ? ?? (resp., ? ? λ ? ?λ ) has no zeros on Σλ (resp., Σ? ). Then Σ is ?nite and, furthermore, ?r ?r Ind ( ?2f ?? 2 , 0) = 2 + #{p ∈ Σλ , λ ? ? ? ?r < 0} ?z ?? ? #{p ∈ Σλ , λ ? ? ? > 0}, ?r ?λ ?2f 2 , 0) = 2 + #{p ∈ Σ? , ? ? λ ? ?r > 0} ?z ?λ < 0}. ? #{p ∈ Σ? , ? ? λ ? ?r

?? ?r

2

Ind (

The hypothesis on the zeros of λ ? ? ?

and ? ? λ ?

?λ ?r

is satis?ed generically (in the

2

C 3 topology, near T , away from the singularity). In particular, since the index of ? f remains ?z 2 2 constant under small C perturbations of f near T , Loewner’s conjecture holds in the following special case: Corollary 1. Let f ∈ C 2 (B, R) be C 3 near T , on Σ? , then Ind

2 ( ? f , 0) ?z 2

?2f ?z 2

= 0 for z = 0. If

?? ?r

≤ 0 on Σλ or

?λ ?r

≥0

≤ 2.

The main idea in the proof of Theorem 1 is to apply Bendixson’s index formula for two2 dimensional singularities to some carefully chosen vector ?elds closely associated to ? f . ?z 2 In the absence of a proof of the full conjecture, the result below has the advantage that it applies under very general conditions. Corollary 2. If f is as in Theorem 1, then Ind ( ?2f , 0) ≤ 2 + min{|Σλ |, |Σ? |}. ?z 2

Acknowledgments. We would like to thank Professors D. Gromoll and B. Lawson for an invitation to spend our last sabbatical year at Stony Brook, where we began to work on this project.

An index formula for Loewner vector ?elds

3

§2 Proof of Theorem 1.

|z|2 2 ,

Replacing f by f ?

f, · , we may assume that

f = 0. Let g(z) =

so that

g(z) = z, Hg (z) = I. Since ?2f ?2 f = 2 (f + cg), ?z 2 ?z we may replace f by f +cg , c > 0 large, and assume further that Hf > 0 and 0 is the only critical point of f in B. It is also easy to see that for large c > 0 the vector ?eld (f + cg) is homotopic to c g through a linear homotopy that vanishes only at z = 0. In particular, renaming f + cg to be f , we may assume Ind ( f, 0) = Ind ( g, 0) = 1. Consider the Cauchy-Riemann operator and its square 1 ? ? ?2 1 ?2 ?2 ?2 ? + i ), ). = ( = ( 2 ? 2 + 2i 2 ?x ?y 4 ?x ?y ?x?y ?z 2 ?z One can identify a symmetric traceless 2 × 2 matrix (aij ) with the complex number a11 + ia12 , and so

(1)

?f f ?2 , = ?z

(Hf ?

?f ? ? 2 f ?f ? 2 f ?f I) = 2 2 , (Hf ? I) f ? 4 2 . = 2 2 ?z ?z ?z

?f 2 ):

In view of our normalizations, the identity below is meaningful (we set h = (2) Ind(

? 2 f ?f ?2f ? 2 f ?f ?f , 0) = Ind( 2 | |2 , 0) = Ind( 2 , 0) + Ind( , 0) = Ind((Hf ? hI) f, 0) + 1. 2 ?z ?z ?z ?z ?z ?z

Consider the vector ?elds (J = the complex structure) (3) X = JHf g, Y = J(Hf ? hI) g.

Again, renaming f + cg (with c > 0 large) to be f , we may assume (4) Ind(J(Hf ? hI) f, 0) = Ind(J(Hf ? hI) g, 0) = Ind(Y, 0).

From (2) and (4) one sees that the task of ?nding a formula for the Loewner vector ?eld is reduced to the computation of Ind(Y, 0).

4

Frederico Xavier

The points in Σλ and Σ? will be referred to as λ-points and ?-points, respectively. Notice that

Hf g = λ g, Hf J g = ?J g at a λ-point, and Hf g = ? g, Hf J g = λJ g at a ?-point. The following observation is fundamental for what follows : (?) The subset of T where either X or Y is tangent to T is precisely the set Σ. We recall Bendixson’s formula for computing the index of an isolated zero p of a smooth planar ?eld ξ ([5], p.173, thm. 9.2 ). Let C be a smooth positively oriented closed curve bounding an open Jordan domain D, p ∈ D. Assume that ξ is de?ned on a neighborhood of D and that p is its only zero. Suppose that ξ is tangent to C at only ?nitely many points. A point q ∈ C where ξ is tangent to C will be called ξ- elliptic (resp., ξ- hyperbolic) if there exists > 0 such that the trajectory of ξ passing through q at time zero is contained in D (resp., contained in the complement of D) for all times t with 0 < |t| < . Bendixson’s index formula asserts that e?h , 2

(5)

Ind(ξ, p) = 1 +

where e and h stand for the number of points in C that are ξ-elliptic and ξ-hyperbolic, respectively. We want to apply (5) to both X and Y , with the choice C = T . Once again, notice the crucial fact that, by (?), the tangencies of X and Y along T occur on the same set. This will enable us to express the index of the vector ?eld Y in terms of quantities associated to X. Let eX , hX , eY , hY denote the number of eliptic and hyperbolic points along Σ, relative to X and Y . We re?ne matters further by considering λ-points and ?-points. Let then eλ and X hλ stand for the number of X-elliptic and X-hyperbolic points that are also λ-points. One can X introduce in a similar fashion the quantities e? , h? , as well as eλ , hλ and e? , h? . The following Y Y X X Y Y relations hold trivially: eX = eλ + e? X X (6) hX = hλ + h? X X eY = eλ + e? Y Y hY = hλ + h? . Y Y

An index formula for Loewner vector ?elds

5

The relations below, on the other hand, are not obvious: eλ + e? = hλ + h? X X X X eλ = hλ X Y (7) hλ = eλ X Y e? = e? X Y h? = h? . X Y The ?rst equation in (7) follows from (5) and (6), since Ind(X, 0) = 1 (this is so because Hf can be deformed into the identity through positive - hence invertible - operators). The proof of the other relations in (7) will involve several computations. The question of whether a particular point is elliptic or hyperbolic, relative to X or Y , will be answered by an examination of how the levels of the function g change along the appropriate trajectory. Clearly, g increases or decreases for small positive times, along a trajectory that is tangent to T at time zero, if and only if the tangency point is hyperbolic or elliptic. Since the ?rst derivative vanishes at time zero – because of tangency –, one must compute second derivatives. We begin our analysis by considering a local trajectory α of X = JHf g that is tangent to T , so that α = X(α), α(0) = p ∈ Σ. We compute

(8)

d2 g(α(t)) = g, JHf g = ? J g, Hf g = dt2 ? JHg X, Hf g ? D3 f (J g, X, g) ? J g, Hf Hg X . g, X = X, g, X .

Remark: For future reference, note that (g ? α) = X Setting A(p) =

d2 g(α(t))|t=0 dt2

and taking p = pλ to be a λ-point in (8) we have, since X(pλ ) = λJ g(pλ ), A(pλ ) = ? ?λ g, λ g ? λD3 f (J g, J g, g) ? J g, λ?J g . Since | g| = 1 on T one has, at pλ , (9) A(pλ ) = λ ? ? ? D3 f (J g, J g, g). λ

Similarly, at a ?-point p = p? we have, using X(p? ) = ?J g, A(p? ) = ? ?? g, ? g ? ?D3 f (J g, J g, g) ? J g, λ?J g ,

6

Frederico Xavier

and so, at p? ,

(10)

A(p? ) = ? ? λ ? D3 f (J g, J g, g). ?

The function Hf J g, J g ? ?| g|2 , de?ned on a neighborhood of B, is non-negative and vanishes precisely at those points where Hf g = λ g. Hence any λ-point pλ is an interior critical point for the function in question, and we have (at pλ ):

0 = ( g) Hf J g, J g ? ?| g|2 = D3 f (J g, J g, g) + Hf J g, JHg g + Hf JHg g, J g ? | g|2 In particular, (11) D3 f (J g, J g, g)(pλ ) = g, ? (pλ ) = ?? (pλ ). ?r g, ? ? 2? Hg g, g .

A similar computation with the non-positive function Hf J g, J g ? λ| g|2 yields, at a ?-point p? , (12) D3 f (J g, J g, g)(p? ) = g, λ (p? ) = ?λ (p? ). ?r

Using (11) and (12), (9) and (10) can be rewritten as (13) A(pλ ) ?? = (λ ? ? ? )(pλ ), λ(pλ ) ?r

(14)

?λ A(p? ) = (? ? λ ? )(p? ). ?(p? ) ?r

We carry out a similar analysis for the vector ?eld Y = J(Hf ? hI) g. Consider a local trajectory β of Y that is tangent to T , so that β = Y (β), β(0) = p ∈ Σ. We compute

(15)

d2 g(β(t)) = g, J(Hf ? hI) g = ? J g, Hf g = dt2 ? JHg Y, Hf g ? D3 f (J g, Y, g) ? J g, Hf Hg Y

An index formula for Loewner vector ?elds

7

If p = pλ is a λ point, or if p = p? is a ?-point, (16) Y (pλ ) = (λ ? h)(pλ )J g(pλ ), Hf Y (pλ ) = ?(λ ? h)(pλ )J g(pλ ), Y (p? ) = (? ? h)(p? )J g(p? ), Hf Y (p? ) = λ(? ? h)(p? )J g(p? ).

Setting d2 B(p) = 2 g(β(t))|t=0 dt one computes, using (15) and (16), B(pλ ) = ? J(λ ? h)J g, λ g ? (λ ? h)D3 f (J g, J g, g) ? J g, ?(λ ? h)J g , so that B(pλ ) = (λ ? h)(λ ? ? ? D3 f (J g, J g, g)). Comparing the last expression with (9), (17) B(pλ ) = (λ ? h) A(pλ ) . λ

It follows from (13), (17), Hf > 0, h = λ+? , and the hypotheses of the theorem that B(pλ ) 2 and A(pλ ) are non-zero and have the same sign. In particular, if pλ is X- hyperbolic (resp., Xelliptic), pλ will be Y - hyperbolic (resp., Y - elliptic). This gives the relations hλ = hλ , eλ = eλ X Y X Y in (7). Finiteness of Σ – and hence the ?niteness of eλ , hλ , e? , h? , eλ , hλ , e? , h? –, will be Y Y X X Y Y X X established shortly. From (15) and (16) we evaluate at p? , B(p? ) = ? J(? ? h)J g, ? g ? (? ? h)D3 f (J g, J g, g) ? J g, λ(? ? h)J g , so that, B(p? ) = (? ? h)((? ? λ) ? D3 f (JDg, J g, g)). Comparing with (10), (18) B(p? ) = (? ? h) A(p? ) . ?

In particular, by (14), (18), Hf > 0 and the hypotheses of the theorem, B(p? ) and A(p? ) are non-zero and have opposite signs. Hence, if p? is X- hyperbolic (resp., X-elliptic), p? will be Y -elliptic (resp., Y -hyperbolic). This gives the remaining relations e? = h? , h? = e? in (7). X Y X Y As mentioned before, we still have to show that Σ is ?nite. The fact that both second order derivatives A(pλ ) and A(p? ) are non-zero means that the X-trajectories that are tangent to the unit circle have simple tangencies (as opposed to a contact of higher order). Using the compactness of T one can then argue geometrically that the set of tangencies, i.e. Σ, must be

8

Frederico Xavier

discrete, as desired. But instead of relying on geometric arguments, we give a short analytic proof that Σ is ?nite. Consider a point p ∈ Σ and the map F (q) = (g(q), X, g (q)),

1 de?ned for all q su?ciently close to p. Observe that F (q) = ( 2 , 0) holds if and only if q lies in T and is a point where X and T are tangent. We want to show that, in a su?ciently small neighborhood of p in R2 , the only such point is q = p. Once this is done, compactness of T implies that Σ is ?nite. By the inverse function theorem, it su?ces to show that DF (p) is invertible. Writing (aij ) for the jacobian matrix of F relative to the oriented basis {e1 = JX(p), e2 = X(p)}, we compute a11 = JX(p), g(p) = ? Hf g, g ,

so that a11 (p) has the value ?λ or ??, according to whether p is a λ-point or a ?-point. In particular, a11 (p) = 0. Also, a21 (p) = X, g (p) = JHf g, g (p) = 0,

because g(p) is an eigenvector of Hf (p). Hence det DF (p) = a11 a22 . It remains to check that a22 (p) = 0. But the quantity a22 (p) = X, X, g (p) was computed already. Indeed, by the remark following (8), a22 (p) is simply A(p) which, under the assumptions of the theorem, was found to be non-zero whether the point p is a λ-point or a ?-point. Hence a22 (p) = 0 and, as explained before, this is enough to show that Σ is ?nite. We can now ?nish the proof of the theorem. From (5), (19) By (6) and (7), eY ? hY = eλ + e? ? hλ ? h? = Y Y Y Y (20) eλ + h? ? hλ ? e? = X X X X eλ + h? ? hλ ? (hλ + h? ? eλ ) = X X X X X X 2(eλ ? hλ ). X X Similarly, eY ? hY = eλ + e? ? hλ ? h? = Y Y Y Y (21) eλ + h? ? hλ ? e? = X X X X (hλ + h? ? e? ) + h? ? hλ ? e? = X X X X X X 2(h? ? e? ). X X Ind(Y, 0) = 1 + eY ? hY . 2

An index formula for Loewner vector ?elds

9

From (2)-(4) and (19)-(21), we have (22) Ind( ?2f , 0) = 2 + eλ ? hλ = 2 + h? ? e? . X X X X ?z 2

In order to identify (22) with the formulas given in the statement of the theorem we appeal to (13) and (14). Indeed, one only needs to recall that λ ≥ ? > 0, a λ-point pλ is X-elliptic (resp., X-hyperbolic) if and only if A(pλ ) < 0 (resp., A(pλ ) > 0), and a ?-point p? is X-elliptic (resp., X-hyperbolic) if and only if A(p? ) < 0 (resp., A(p? ) > 0).

References

[1] [2]

¨ Bol, G., Uber Nabelpunkte auf einer Ei?¨che, Math. Z. 49 (1943-1944), 389-410. a Gutierrez. C, Sotomayor, J., Structurally stable con?gurations of lines of curvature, Asterisque 98-99 (1982), 195-215. Hamburger, H.L., Beweis einer Carath?odoryschen Verm¨tung. I, Ann. of Math. 41 (1940), 63-86. e u Hamburger, H.L., Beweis einer Carath?odoryschen Verm¨tung. II, III, Acta Math. 73, (1941), 174-332. e u Hartman, P., Ordinary Di?erential equations, John Wiley, New York, 1964. Ivanov, V.V., The analytic Carath?odory Conjecture, Siberian Math. J. 43 (2002), 251-322. e Klotz, T., On G. Bol’s proof of Carath?odory’s conjecture, Comm. Pure Appl. Math. 12 (1959), 277-311. e Lazarovici. L., Elliptic sectors in surface theory and the Carath?odory-Loewner conjectures, J. Di?. Geom. e 55 (2000), 453-473. Smyth, B, Xavier, F., A sharp geometric estimate for the index of an umbilic on a smooth surface, Bull. London Math. Soc. 24 (1992), 176-180.

2 Smyth, B., Xavier, F., Real solvability of the equation ?z f = g and the topology of isolated umbilics, J. Geom. Anal. 8 (1998), 655-671.

[3] [4] [5] [6] [7] [8]

[9]

[10]

[11]

Titus, C., A proof of a conjecture of Loewner and of the conjecture of Carath?odory on umbilic points, e Acta Mat. 131 (1973), 43-77.

Frederico Xavier Department of Mathematics University of Notre Dame, Notre Dame IN, 46635 email xavier.1@nd.edu

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