Planar vector field versions of Carathéodory's and Loewner's conjectures

Publicacions Matem`atiques, Vol 41 (1997), 169–179. PLANAR VECTOR FIELD VERSIONS OF CARATH´ EODORY’S AND LOEWNER’S CONJECTURES Carlos Gutierrez and Federico S´ anchez-Bringas Abstract Let r=3,4,... ,∞,ω.TheCr-Carath´eodory’s Conjecture states that every Crconvex embedding of a 2-sphere into R3must have at least two umbilics. The Cr-Loewner’s conjecture (stronger than the one of Carath´eodory) states that there are no umbilics of index bigger than one. We show that these two conjectures are equivalent to others about planar vector fields. For instance, if r=ω, Cr-Carath´eodory’s Conjecture is equivalent to the following one: Let ρ>0 and β:U⊂R2→R, be of class Cr, where U is a neighborhood of the compact disc D(0,ρ)⊂R2of radius ρcentered at 0. If βrestricted to a neighborhood of the circle ∂D(0,ρ) has the form β(x, y)=(ax2+by2)/(x2+y2), where a<b<0, then the vector field (defined in U) that takes (x, y)to (βxx(x, y)−βyy (x, y),2βxy (x, y)) has at least two singularities in D(0,ρ). 1. Introduction The classical Carath´eodory’s Conjecture states that every smooth convex embedding of a 2-sphere in R3, i.e. an ovaloid, must have at least two umbilics. A well known approach to the problem is based on a “semilocal” argument. For any surface in R3, the eigenspaces of the second fundamental form define two orthogonal line fields (principal directions) whose singularities are exactly the umbilics. To each isolated umbilic we can attach the index of either one of the two fields, which is half of an integer, and the sum of these indexes is the Euler characteristic of the surface, if this is compact and all umbilics are isolated. So, if an ovaloid Supported in part by Proyecto D.G.A.P.A. IN 103795 UNAM and by IMPA-Rio de Janeiro. 170 C. Gutierrez, F. S´ anchez-Bringas has only one umbilic, it must have index two. We just observe that, up to an inversion in R3, we can always suppose that the curvature at a given umbilic is positive and therefore the convexity hypothesis is not relevant for this argument. Examples of umbilics of index jare known for all j≤1 and a local conjecture, known as the Loewner conjecture, states that there are no umbilics of index bigger than one. Several authors, among whom are H. Hamburger [Ham], G. Bol [Bol], T. Klotz [Klo], C. J. Titus [Tit] and Scherbel [Sch] have asserted this conjecture for analytic surfaces to be true; implying therefore Carath´eodory’s Conjecture for analytic surfaces. Nevertheless, Klotz [Klo] pointed out a gap in Bol’s proof; also, Scherbel [Sch] claims that there are gaps in the works of Klotz and Titus (see also [Lan], [Yau]). Related to the subject, we wish to mention the works [GMS] of Gutierrez, Mercuri and S´anchez-Bringas, [GS1], [GS2] of Gutierrez and S´anchez-Bringas, and [SX1], [SX2] of Smyth and Xavier. In this paper we show that these two conjectures are equivalent to others about planar vector fields. We hope that our results help to obtain simpler solutions in the analytic case and to find ways to attack the Crcase. We wish to thank the referee, whose comments were appreciated and incorporated into this work. 2. Preliminaries Orient the sphere S2⊂R3so that the positive unitary normal vector at p∈S2is pitself. Let S⊂R3beaCr-ovaloid. By this we mean that Sis an oriented Cr-embedded surface such that its Gauss map N:S→S2is an orientation preserving diffeomorphism. This definition implies that Sis convex, compact and that its Gaussian curvature is positive everywhere. We define the support function of Sas the map σ:S2→Rgiven by σ(p)=p·N−1(p), where the dot stands for the usual inner product. Given δ∈{−,+}, let Πδ:R2→S2\{(0,0,δ(1))}be the diffeomorphism given by Πδ(x, y)=2x 1+x2+y2,2y 1+x2+y2,δ(x2+y2−1) 1+x2+y2. That is, Πδis the inverse map of the corresponding stereographic projection. The map Φδ(x, y)=(Xδ(x, y),Yδ(x, y),Zδ(x, y)) = N−1◦Πδ(x, y), On Carath´ eodory’s and Loewner’s Conjectures 171 defined in R2 , provides a global Cr−1parametrization of S\{N−1 (0,0,δ(1))} called Bonnet chart associated to (S,Πδ). Given the support function σ of S, associated to (S,Πδ) we define the Bonnet function βδ(x, y) = (1 + x2+y2)σ(Πδ(x, y)). Let Λδ(x, y) = (1 + x2+y2)Πδ(x, y); that is, Λδ(x, y)=(2x, 2y,δ(x2+y2−1)). As Λδ·Φδ x=Λ δ·Φδ y= 0, where the subindex means the partial derivative with respect to this variable, we have that Λδ x·Φδ=βδ xand Λδ y·Φδ=βδ y. This together with Λδ·Φδ=βδcan be written in matrix notation as Mδ·Φδ=Bδ, where (1) Mδ=  2x2yδ(x2+y2−1) 20 δ(2x) 02 δ(2y)  ,Φδ=  Xδ Yδ Zδ ,Bδ=  βδ βδ x βδ y  . As Nis of class Cr−1,Φ δis also of class Cr−1. Therefore, Mδ·Φδ=Bδ implies that βδis of class Cr. Since, for all (x, y)∈R2, the determinant of Mδis −(δ)4(1 + x2+y2)= 0, we may write Φδ=(Mδ)−1·B δ. From this, using a symbolic computer system we can obtain the first and second fundamental forms of Φδand therefore the proof of proposition below (see [GMS], [Dar], [Bon]). Proposition 2.1. Let S⊂R3be a Crovaloid, r≥3. Then the support function σof Sis of class Crand the differential equation of the principal lines of curvature of Sin its Bonnet chart, associated to (S,Πδ), is given by ωδ=0where (2) ωδ=βδ xydx2+(βδ yy −βδ xx)dxdy −βδ xydy2 and βδ(x, y) = (1 + x2+y2)σ(Πδ(x, y)). The proof of the following proposition and theorem can be found in [GMS]. Proposition 2.2. Let β:R2→Rbe a Crfunction, with r≥3. Suppose that the 2-jet j2β(0,0) of βat (0,0) has the form (3) j2β(0,0)(x, y)=a00 +a10x+a01y+a20x2+2a11xy +a02y2 and that (4) a2 00 +a00(a02 +a20)−a2 11 +a02a20 =0. Then there exists an open neighborhood U⊂R2of (0,0) such that β|U is the Bonnet function of an oriented Crsurface embedded in R3. 172 C. Gutierrez, F. S´ anchez-Bringas The following will be needed later Lemma 2.3. Let δ∈{−,+}and let βδ(x, y) = (1 + x2+y2)σ(Πδ(x, y)) be the functions determined by σas above. If I:2\{0}→ 2\{0}is the inversion I(u, v)=x x2+y2,y x2+y2, then, for all (x, y)∈ 2\{0}, (5) (x2+y2)β−◦I(x, y)=β+(x, y). Proof: The result follows inmediately from the identity σ(Π+(x, y)=σ(Π−◦I(x, y)). Let Fbe a one dimensional C1-foliation defined on a neighborhood U of 0 ∈R2. Suppose that Fhas exactly one singularity which is 0. The index of 0 (i.e. of Fat 0) is one-half of the degree of the map that takes each point qof a small circle centered at 0 to the element, of the projective circle RP, which is tangent at qto the leaf of F. Here, we identify R2with the tangent space of Uat q; also, the projective circle RPis the well known quotient space obtained from R2\{0}.IfFis orientable, this definition coincides with the usual Hopf-Poincar´e index. If pis an isolated umbilic point of a Croriented surface S⊂R3, with r≥3, the index of pis defined by using local coordinates and either one of the two foliations induced by the principal lines of curvature of S. The umbilics are precisely the singularities of these foliations. We say that a Crvector field ξon R2fulfills a Lojasiewicz-inequality at (0,0) if there exist k∈N∗and δ>0 such that ||ξ(x, y)|| ≥ δ||(x, y)||k on some neighborhood of (0,0). Under these circumstances, we will also say that ξsatisfies a Lojasiewicz-inequality of order k(with associated constant δ)at(0,0). Suppose that a Croriented surface S⊂R3, with r≥3, has an isolated umbilic point p∈S. We will say that pis an umbilic of Lojasiewicz-type (of order kwith 1 ≤k≤r−2) if there is a local Crdiffeomorphism ϕof a neigborhood of p∈R3onto an open set of R3such that the image surface ϕ(S)= ˜ Ssatisfies the following properties: a) ˜p=ϕ(p) is an isolated umbilic of ˜ Swith the same index of pand ˜ Shas positive curvature in ˜pand unit normal vector (0,0,−1). b) The Bonnet function ˜ βof ˜ Sis such that the vector field ˜ ξ(x, y)= (˜ βxx −˜ βyy,2˜ βxy) satisfies a Lojasiewicz-inequality of order kat (0,0). On Carath´ eodory’s and Loewner’s Conjectures 173 Remark 2.4. (a) The composition of an appropriate rigid translation and the inversion I(p)= p ||p||2preserves the principal lines of curvature, hence umbilics and their indexes as well. Thus an inversion may be used to transform a flat umbilic into an umbilic of positive curvature. Therefore, up to a conformal diffeomorphism, the first condition is always satisfied. (b) With the notation right above, the index of ξat (0,0) is twice the index of the umbilic point p[SX1]. (c) If a vector field on R2satisfies a Lojasiewicz-inequality at the singular point (0,0), then (0,0) is an isolated singularity of the vector field. (d) Suppose that Y:(U, (0,0)) →(R2,(0,0)) is an analytic vector field defined in an open set U⊂R2. Then (0,0) is an isolated singular point of Yif, and only if, Ysatisfies a Lojasiewicz-inequality at (0,0) [Loj]. Therefore, using (a) above, an analytic surface immersed in R3always satisfies a Lojasiewiecz-inequality at an isolated umbilic point. The proofs of the following lemma and theorem right below are given in ([GMS]). Lemma 2.5. If ξ:(U, (0,0)) →(R2,(0,0)) is a Crvector field, r≥1, defined in a neighborhood Uof (0,0) and satisfying a Lojasiewiczinequality of order k,1≤k≤rat (0,0), then (a) the k-jet jkξ0of ξat (0,0) satisfies a Lojasiewicz-inequality of order kat (0,0) (b) both ξand its k-jet jkξ0at (0,0) have the same index at their common isolated singularity (0,0). Theorem 2.6. Assuming the truth of the Loewner’s Conjecture for isolated umbilics on analytic surfaces, if a Crsurface S⊂R3, with r≥3, satisfies a Lojasiewicz-inequality at an umbilic point p, then the index of pis at most 1. Therefore if a Crimmersion of a sphere has one umbilic of Lojasiewicz type, it must have at least one more umbilic. A proof of the following result can be found in [GS1]. See also [D-G], [Fir], [LLR], [Nir]. 174 C. Gutierrez, F. S´ anchez-Bringas Theorem 2.7. Let S⊂R3beaCrovaloid, r≥3. Then the inverse N−1:S2→S, of the Gauss map N, can be written as follows: N−1(u, v, w)=σ(u, v, w)·(u, v, w)+A(u, v, w)·∇σ(u, v, w) where σ:S2→Rdenotes the support function of S,∇σits gradient vector field and (6) A(u, v, w)=  v2+w2−uv −uw −uv u2+w2−vw −uw −vw u2+v2 . Conversely, given a Crfunction σ:S2→R,r≥3, there exists a constant c>0such that σ+cis the support function of an ovaloid of class Cr. 3. Equivalent conjectures Let r=3,4,... ,∞,ω. The conjectures that we are interested in are the following ones: Cr-Loewner’s Conjecture (i.e. Cr-LC). The index of an umbilic, of a surface Crembedded in R3, is at most one. Cr-Loewner’s Conjecture* (i.e. Cr-LC*). Let β:U⊂R2→R, be a map of class Crdefined in a neighborhood U of (0,0) ∈R2.If(0,0) is an isolated singularity of the vector field X:(x, y)→(βxx −βyy,2βxy), then the index of Xat (0,0) is less or equal than 2. Cr-Loewner’s Conjecture with Lojasiewicz condition (i.e. CrLC with LC). The index of an umbilic of Lojasiewicz-type, of a surface Crembedded in R3, is at most one. Cr-Loewner’s Conjecture* with Lojasiewicz condition (i.e. Cr-LC* with LC). Let βand Xbe as in Cr-LC*. If Xsatisfies a Lojasiewicz Condition at the singularity 0, then the index of Xat 0 is less or equal than 2. On Carath´ eodory’s and Loewner’s Conjectures 175 Cr-Carath´eodory’s Conjecture (i.e. Cr-CC). Every Crconvex embedding of a 2-sphere in R3must have at least two umbilics. Cr-Carath´eodory’s Conjecture* (i.e. Cr-CC*). Let ρ>0 and β:U⊂R2→R, be of class Cr, where Uis a neighborhood of the compact disc D(0,ρ)⊂R2of radius ρcentered at 0. If βrestricted to a neighborhood of the circle ∂D(0,ρ) has the form β(x, y)=ax2+by2 x2+y2 where a<b<0, then the vector field (defined in U) X:(x, y)→(βxx −βyy,2βxy) has at least two singularities in D(0,ρ). Theorem 3.1. Let r=3,4,... ,∞,ω. (a) Cr-LC is equivalent to Cr-LC* (b) Cr-LC with LC is equivalent to Cr-LC* with LC (c) The following conjectures are equivalent: (c1) Cω-LC*, (c2) Polynomial-LC* (c3) Cr-LC* with LC. (d) If r=ω,Cr-CC is equivalent to Cr-CC* Proof: The proofs of (a) and (b) are the same; they follow from Propositions 2.1, 2.2 and Remark 2.4(b). The proof of (c) follows from Lemma 2.5 and Remark 2.4(d). See [GMS, Theorem 3.3]. Let us proceed to prove (d). We shall use the notations introduced in Section 2. First, we shall see that if Cr-CC* is true then Cr-CC is true. Without lost of generality, we may assume that the ovaloid is tangent to the xy-plane at ¯ 0=(0,0,0), that ¯ 0 is not an umbilic point and that N(¯ 0) = (0,0,1). By using the formula M·Φ−=B−and the assumption that Φ(0,0) = ¯ 0, we conclude that β−(0,0) = β− x(0,0) = β− y(0,0) = 0. Therefore, by rotating the ovaloid around the z-axis if necessary, we may assume that β−(x, y)=ax2+by2+ higher order terms, 176 C. Gutierrez, F. S´ anchez-Bringas and that |a|≤|b|. It is easy to see that the assumptions imply that β has a local maximun at ¯ 0. By the convexity of the ovaloid and as ¯ 0is not umbilic, we obtain that a<b<0. By a small C2-perturbation of the ovaloid, around ¯ 0, we may assume that, around ¯ 0, β−(x, y)=ax2+by2. As, by Lemma 2.3, β+(u, v)=(u2+v2)β−u u2+v2,v u2+v2 we get that, there exists ρ>0 such that, for all (u, v) in an neighborhood of {(u, v):u2+v2=ρ2}, β+(u, v)=au2+bv2 u2+v2. By the assumption this implies that the vector field (u, v)→(β+ uu −β+ vv,2β+ uv) has at least two singularities in {(u, v):u2+v2≤ρ2}. Each of these singularities is taken by the parametrization Φ+to an umbilic of the ovaloid. This proves that Cr-CC is true. Now, we shall see that if Cr-CC is true then Cr-CC* is true. Suppose that we have a function β=β+=β+(u, v), and real numbers ρ,a,b, with a<b<0, as in the assumptions of Cr-CC*. In particular, we are assuming that β+restricted to a neighborhood of the circle ∂D(0,ρ) has the form β+(u, v)=au2+bv2 u2+v2. Without lost of generality, we may assume that β+is defined in the whole R2and that, when restricted to a neighborhood of the set {(u, v)∈R2:u2+v2≥ρ}, is given by the expression right above. Define β−=β−(x, y), in R2\{(0,0)}by the expression β−(x, y)=(x2+y2)β+x x2+y2,y x2+y2. Let β−(0,0) = 0. It can be seen that β−restricted to a neighborhood of S=(x, y):x2+y2≤1 ρ2 On Carath´ eodory’s and Loewner’s Conjectures 177 has the form β−(x, y)=ax2+by2. By construction, β+and β−can be made the Bonnet functions of the support function σ:S2→Rof class Cr. By Theorem 2.8, there exists c∈Rsuch that we have an ovaloid associated to σ+c. By the assumptions, the ovaloid has at least two umbilics. These umbilics must be contained in Φ+(D(0,ρ)). Each of these umbilics is taken by (Φ+)−1to a singularity (contained in D(0,ρ)) of the vector field (u, v)→(β+ uu −β+ vv,2β+ uv). This proves that if Cr-CC is true then Cr-CC* is also true. Remark 3.2. Let S⊂R3beaCrovaloid and N−1:S2→Sbe inverse of the Gauss map N. There exists a quadratic differential form ω on S2, as defined by V. Gu´ı˜nez (see [Gui], [Mic]), such that its expression in the parametrization Πδ, with δ∈{−,+}, is the quadratic differential ωδof Proposition 2.1. The map N−1takes the pair of foliations associated to ωto the pair of foliations tangent to the principal lines of S. The foliations associated to ωδ, under the name of Hessian foliations, are considered in [SX2]. References [Bon] O. Bonnet,M´emoire sur l’emploi d’un nouveau syst`eme de variables dans l’´etude des propri´et´es des surfaces courbes, J. Liouville, Ser. 2 5(1860), 153–266. [Bol] G. Bol,¨ Uber Nabelpunkte auf einer Eifl¨ache, Math Z. 49 (1943/1944), 389–410. [D-G] M. Dajczer and D. Gromoll, Gauss parametrizations and rigidity aspects of submanifolds, J. Differential Geom. 22 (1985), 1–12. [Dar] G. Darboux, Sur la forme des lignes de courbure dans la voisinage d’un ombilic, Le¸cons sur la Theorie des Surfaces, IV, Note 7, Gauthier Villars, Paris (1896). [Gui] V. Gu´ ı˜ nez, Locally stable singularities for positive quadratic differential forms, J. Differential Equations 110 (1994), 1–37. [GMS] C. Gutierrez, F. Mercury and F. S´ anchez-Bringas, On a Carath´eodory’s Conjecture: analyticity versus smoothness, Experiment. Math. 5(1) (1996), 33–38.