Structurally stable configurations of lines of mean curvature and umbilic points on surfaces immersed in R3

Publ. Mat. 45 (2001), 431–466 STRUCTURALLY STABLE CONFIGURATIONS OF LINES OF MEAN CURVATURE AND UMBILIC POINTS ON SURFACES IMMERSED IN R3 Ronaldo Garcia and Jorge Sotomayor Abstract In this paper we study the pairs of orthogonal foliations on oriented surfaces immersed in R3whose singularities and leaves are, respectively, the umbilic points and the lines of normal mean curvature of the immersion. Along these lines the immersions bend in R3according to their normal mean curvature. By analogy with the closely related Principal Curvature Configurations studied in [S-G], [GS2], whose lines produce the extremal normal curvature for the immersion, the pair of foliations by lines of normal mean curvature and umbilics, assembled together, are called Mean Curvature Configurations. This paper studies the stable and generic cases of umbilic points and mean curvature cycles, with their Poincar´e map. This provides two of the essential local ingredients to establish sufficient conditions for mean curvature structural stability, the analog of principal curvature structural stability,[S-G], [GS2]. 1. Introduction In this paper are studied the Mean Curvature Configurations associated to immersions of an oriented surface into R3. They consist on the Umbilic Points and the Lines of Normal Mean Curvature of the immersion. These lines are characterized by the condition that along them the immersions bend according to their Normal Mean Curvature. There are many classical pictorial examples and analytic properties of Principal Configurations. Such are the cases of surfaces which are quadratic, of revolution, normally translated, triply orthogonal and inverted, 2000 Mathematics Subject Classification. 53A07, 53A05. Key words. Umbilic point, mean curvature configuration, lines of mean curvature. The first author was partially supported by CNPq and FAPESP, Grant 99/08487-7. The second author was partially supported by CNPq. This work was done under the project PRONEX/FINEP/MCT - Conv. 76.97.1080.00 - Teoria Qualitativa das Equa¸c˜oes Diferenciais Ordin´arias. 432 R. Garcia, J. Sotomayor among other types. However, the Mean Curvature Configurations do not seem to have been considered before, even for quadrics. An exception is the classical result that on Minimal Surfaces it is given by the Configuration of Asymptotic Lines [St]. In order to illustrate the difference, in Section 5 will be studied the specific examples of ellipsoids and tori, showing that in these cases where the principal lines are trivially periodic, the mean curvature ones may be dense. One of the sources of interest for this work is the formal analogy with previous work on Principal Configurations [S-G], [GS1]–[GS3], dealing with the umbilic points and integral foliations of tangent principal direction fields along which the immersions bend producing the extremal (principal) normal curvature. In fact, by Euler’s Formula, [St], the directions of extremal (i.e. principal) geodesic torsion coincide with those of normal mean curvature. Also, problems of singularities and global analysis as the Carath´eodory conjecture about the number of umbilics on smooth ovaloids is a source for the study of mean curvature lines and principal lines, [Gu], [Ga-M-F-R], [SX1] and [SX2]. It will clear that the specific results suggested by the formal analogy between the Mean and Principal Configurations, when formulated precisely, need to be proved again. When necessary, this will be done here. See for instance Section 3 where the mean curvature separatrix structures at umbilic points, analogous to the Darbouxian principal curvature configurations [GS2], has been established. See also [Da] and [B-F]. There seems to be no general formal argument, a sort of functor, which enables to establish a result in one setting as a corollary of the one to which it is linked by the analogy. However, the methods used for the proofs have some expected similarity, coming from the analysis of quadratic differential equations which govern both Mean and Principal Curvature Lines on a surface. The ideas formulated above go beyond the formal analogy. This is better explained in terms of the Axial Configuration and the Ellipse of Curvature of immersions into R4,[Lit]. In fact the large axis of the Ellipse of Curvature gives rise naturally to the Principal Curvature Configurations and the Small Axis leads to the Mean Curvature Configurations, as established in [Ga-So1]. See also [G-G-T-G]. Sufficient conditions are provided here to extend to the present Mean Curvature setting the Theorems on Structural Stability for Principal Configurations proved in [S-G], [GS2] and for Axial Structural Stability [Ga-So1]. Lines of Mean Curvature on Surfaces Immersed in R3433 Two local cases are essential for this extension: the umbilic points with the determination of their mean curvature separatrix structure and the mean curvature cycles with the calculation of the derivative of the Poincar´e map. Both are treated in detail here. This will establish that, in this aspect, the parallel with the conditions for principal structural stability is remarkable. This paper is organized as follows: Section 2 is devoted to the general study of the differential equations of Mean Curvature Lines. The precise definition of Mean Curvature Configuration and Mean Curvature Structural Stability will be given at the end of this section. In Section 3 the equation of lines of mean curvature is written in a Monge chart. The umbilic stability condition is explicitly stated in terms of the coefficients of the third order jet of the function which represents the immersion in a Monge chart. The local mean curvature separatrix configurations at stable umbilics is established for C4immersions. In Section 4 the derivative of first return Poincar´e map along a mean curvature cycle is established. It consists of an integral expression involving the curvature functions along the cycle. Section 5 presents examples of Mean Curvature Configurations on the Ellipsoid and the Torus. In Section 6 the results presented in Sections 3 and 4 are put together to provide sufficient conditions for Mean Curvature Structural Stability. Section 7 contains the study of the density of Mean Curvature Structurally Stable immersions. In a sense the present work is intermediate between the studies of Principal and Axial configurations of lines of curvature on a surface. In the forthcoming paper [Ga-So2] the results about Mean and Principal Curvature Structurally Stable Configurations for surfaces immersed into R3will be put together as essential pieces to provide a class of Structurally Stable Axial Configurations of surfaces immersed into R4. This is achieved by means of a suitable deformation of immersions from R3 into R4. In this paper this is only illustrated in the case of immersions of the Torus free of umbilics. The general problem, proposed in [Ga-So1], will be dealt with in [Ga-So2]. Section 8 is devoted to explaining better this point. 2. Differential equation of mean curvature lines Let α:M2→R3beaCr,r≥4, immersion of an oriented smooth surface Minto R3. This last space is oriented by a once for all fixed 434 R. Garcia, J. Sotomayor orientation and endowed with the Euclidean inner product ,. Let N be a vector field orthonormal to α. Assume that (u, v) is a positive chart and that {αu,α v,N}is a positive frame in R3. In the chart (u, v), the first fundamental form of an immersion αis given by: Iα=Dα,Dα=Edu2+2Fdudv +Gdv2, with E=αu,α u, F=αu,α v,G=αv,α v. The second fundamental form is given by: IIα=N,D2α=edu2+2fdudv +gdv2. The normal curvature at a point pin a tangent direction t=[du :dv]is given by: kn=kn(p)=IIα(t, t) Iα(t, t). The geodesic torsion at a point pin a tangent direction [du :dv]is given by: τg=(Fg−Gf)dv2+(Eg −Ge)dudv +(Ef −Fe)du2 (EG −F2)3 2(Edu2+2Fdudv +Gdv2). The lines of mean curvature are defined by the condition of having normal curvature equal to the mean curvature, i.e., kn=H. Therefore the pertinent differential equation is given by: edu2+2fdudv +gdv2 Edu2+2Fdudv +Gdv2=Eg +eG −2fF 2(EG −F2)=H. Or equivalently by [g(EG −2F2)+2fFG−eG2]dv2+[4fEG−2gEF −2eFG]dudv +[e(EG −2F2)+2fEF −gE2]du2=0. This is equivalent to assert that these lines make extremal the geodesic torsion. In fact, τg=±√H2−Kis the extremal value of τg= √H2−Kcos θsin θ, attained in a mean curvature direction θ=π 4[St]. As in the case of principal lines, the two mean curvature directions defined by the equation above are orthogonal with respect to the induced metric. This is immediate by the extremal property of the geodesic torsion. In a minimal surface mean curvature lines coincide with the asymptotic lines [St]. Remark 2.1.The differential equation of mean curvature lines can also be written as Jac(Jac(IIα,I α),I α)=0. Lines of Mean Curvature on Surfaces Immersed in R3435 In Monge chart (u, v, h(u, v)) the differential equation of lines of mean curvature is given by: [hvv(1 + h2 u+h2 v−h2 uh2 v)+2huvhuhv(1 + h2 v)−huu(1 + h2 v)2]dv2 +[4huv(1 + h2 u)(1 + h2 v)−2hvvhuhv(1 + h2 u)−2huuhuhv(1 + h2 v)]dudv +[huu(1 + h2 u+h2 v−h2 uh2 v)+2huvhuhv(1 + h2 u)−hvv(1 + h2 u)2]du2=0. The developments above allow us to organize the lines of mean curvature of immersions into the mean curvature configuration, as follows: Through every point p∈M\U α, where Uαis the umbilic set of α, pass two orthogonal mean curvature lines of α. Under the orientability hypothesis imposed on M, the mean curvature lines define two foliations Qα,1, called the minimal mean curvature foliation, along which the geodesic torsion is minimal (i.e. τg=−√H2−K), and Qα,2, called the maximal mean curvature foliations, along which the geodesic torsion is maximal (i.e. τg=+ √H2−K). These foliations are singular at the umbilic points of α. The triple Qα={Uα,Qα,1,Qα,2}is called the mean curvature configuration of α. An immersion αis said Cs-local mean curvature structurally stable at a compact set K⊂Mif for any sequence of immersions αnconverging to αtogether with its first sderivatives in a compact neighborhood VK of Kthere is a sequence of compact subsets Knand a sequence of homeomorphisms mapping Kto Knconverging to the identity of Msuch that on VKit maps arcs of the mean curvature foliations Qα,i to arcs of Qαn,i for i=1,2. An immersion αis said Cs-mean curvature structurally stable if the compact Kabove is M. In Sections 3 and 4 will be characterized the umbilics and mean curvature cycles which are local mean curvature structurally stable. Sufficient conditions for immersions to be mean curvature structurally stable are given in Section 6. 3. Mean curvature lines near umbilic points Let 0 be an umbilic point of a C4immersion αparametrized in a Monge chart (x, y)byα(x, y)=(x, y, h(x, y)), where h(x, y)=k 2(x2+y2)+a 6x3+b 2xy2+c 6y3+O(4).(3.1) 436 R. Garcia, J. Sotomayor The differential equation of principal curvature lines is given by: −[by +P1]dy2+[(b−a)x+cy +P2]dxdy +[by +P3]dx2=0(3.2) and the differential equation of the mean curvature lines is given by: (3.3) [(b−a)x+cy +M1]dy2 +[4by +M2]dxdy −[(b−a)x+cy +M3]dx2=0 where Miand Pi,i=1,2,3, represent functions of order O((x2+y2)). Let ∆H=4c2(2a−b)2−[3c2+(a−5b)2][3(a−5b)(a−b)+c2] ∆P=4b(a−2b)3−c2(a−2b)2. (3.4) As a starting point, recall the behavior of principal lines near Darbouxian umbilics in the following proposition. Proposition 3.1 ([S-G], [GS2]).Under the conditions above suppose that the transversality condition T=b(b−a)=0holds and consider the following situations: D1)∆ P>0. D2)∆ P<0and a b>1. D3)a b<1. Then each principal foliation has in a neighborhood of 0, one hyperbolic sector in the D1case, one parabolic and one hyperbolic sector in D2case and three hyperbolic sectors in the case D3. The umbilics are called Darbouxians of types D1,D2and D3. Proposition 3.2. Suppose that the transversality condition T=b(b− a)=0holds and consider the following situations: H1)∆ H>0. H2)∆ H<0and a b>1. H3)a b<1. Then each mean curvature foliation has in a neighborhood of 0, one hyperbolic sector in the H1case, one parabolic and one hyperbolic sector in H2case and three hyperbolic sectors in the case H3. These umbilic points are denoted by H1,H2and H3. The mean curvature configuration Qαnear an umbilic point of type Hi has a local behavior as shown in Figure 3.1. Lines of Mean Curvature on Surfaces Immersed in R3437 H1H2H3 Figure 3.1. Mean curvature lines near the umbilic points Hi. Remark 3.1.In the propositions above in D3case (resp. H3case), the condition ∆P<0 (resp. ∆H<0) is automatically satisfied with a b<1. Proof of Proposition 3.2: The proof is inspired on that of Proposition 3.1, for the case of Darbouxian umbilic points; see [B-F] and [GS2]. Consider the implicit surface G(x, y, p)=[(b−a)x+cy +M1(2)]p2 +[4by +M2(2)]p−[(b−a)x+cy +M3(2)] = 0, which, under the transversality condition T=(b−a)b= 0, is regular near the projective line represented by the p-axis i.e. x=0,y=0. In what follows it will be supposed, without loss of generality, that c b>0. The mean curvature lines in the chart (x, y) are the projections by π(x, y, p)=(x, y) of the integral curves of the Lie-Cartan line field X given locally by: X=Gp ∂ ∂x +pGp ∂ ∂y −(Gx+pGy)∂ ∂p. 438 R. Garcia, J. Sotomayor The projection πis a double regular covering outside the projective line and it is singular only along the projective line. By the transversality condition b(b−a)= 0, the singular points of X are contained in the projective line and are given by (0,0,p), where p are the roots of the cubic polynomial R(p)=Gx+pGy|(x,y)=(0,0) =cp3+(5b−a)p2−cp +a−b. The discriminant of the polynomial Ris precisely ∆H, defined in equation (3.3). There are the three cases to consider. Case 1: a b<1. Then the polynomial Rhas three real roots, r1∈(0,1), r2∈(−1,0) and r3∈(−∞,−1). This can be seen intersecting the graphs of the curves l(p)=c b(p3−p) and m(p)=( a b−5)p2+1−a b. The linear part of Xat a singular point (0,0,p) is given by: DX(0,0,p)=  2p(b−a)4b+2c(p2−1) 0 2(b−a)p2(4b+2c(p2−1))p0 ∗∗−R(p) . Therefore the non zero eigenvalues of DX(0,0,p) are given by: λ1(p)=2(b−a)p+(4b+2c(p2−1))pand λ2(p)=−R(p). Direct calculation shows that R|{λ1=0}=2bp +a−band λ1|{R=0}= −4bp(p2+1) p2−1. Then the two polynomials R(p) and λ1(p) have no common roots. So, it is obtained that λ1(r3)>0, λ1(r2)<0 and λ1(r3)>0. As we are assuming that c b>0 and a b<1 it follows that λ2(r3)<0, λ2(r2)>0 and λ2(r3)<0. Therefore, λ1(ri)λ2(ri)<0 for i=1,2,3. The three singular points are hyperbolic saddles. This ends the proof of the H3case. Case 2: a b>1 and ∆H<0. Then the polynomial R(p) has three real roots and these are located as follows: r1,r 2∈(0,1), r2<r 1and r3∈(−∞,−1). As in the situation above, the points (0,0,r 1) and (0,0,r 3) are hyperbolic saddles and (0,0,r 2) is a hyperbolic node. This ends the proof of the H2case. Lines of Mean Curvature on Surfaces Immersed in R3439 Case 3: ∆H>0. Then the cubic polynomial R(p) has only one real root and it follows that this root is located in the interval (−∞,−1) which is a hyperbolic saddle. This ends the proof of the H1case. To conclude the proof we observe that, in the chart (u, v, q), q=du dv = 1 p, the Lie-Cartan line field, Y=qGq ∂ ∂x +Gq ∂ ∂y −(qGx+Gy)∂ ∂q defined from the implicit differential equation G(u, v, q) = 0, is regular, i.e., Y(0) = 0, provided c=0. The case c b<0 can be analyzed similarly. If c= 0, the point (0,0,0), in (x, y, q) coordinates, is a hyperbolic singular point and the analysis of the mean curvature configuration is the same as above. Figure 3.2 illustrates the behavior of Xnear the projective line. Figure 3.2. Phase portrait of Xnear the projective line. Proposition 3.3. In the plane b=1the diagram of the umbilic points of types Hifor the mean curvature configuration and of types Difor the principal configuration is as shown in the Figure 3.3. Proof: The curve ∆P= 0 is the union of a parabola with vertex at (2,0) and the line a= 2, while the curve ∆H= 0 is a curve of order four, having two connect components, with a vertex at (1,0) and a cuspidal pointat(5,0). The transversal intersection between ∆P= 0 and ∆H=0 occurs at the points (2,±9+6 √3) and (11+6√3,±2(9+6 √3). 446 R. Garcia, J. Sotomayor Proposition 5.2. Let αbe an immersion of a surface Minto R3and consider the displacement α=α+9Nα, where Nαis the normal map of α. Then the principal lines are preserved along αwhile the mean curvature lines are not. In fact, the mean curvature lines are locally rotated by a nonzero angle. Proof: Let (u, v) be a principal chart on the surface M. The coefficients of the fundamental forms of αand of αin the principal chart (u, v) are related by: ¯ E=(1−9k1)2E, ¯e=(1−9k1)e ¯ F=F=0,¯ f=f=0 ¯ G=(1−9k2)2G, ¯g=(1−9k2)g. So, the differential equation of mean curvature lines for the immersion α is given by: (1 −9k1)2Edu2−(1 −9k2)2Gdv2= 0. Therefore dv du =±E G 1−k1 1−k2. It follows that d d (dv du )|{=0}=±E G(k2−k1)=0. 5.2. Mean curvature lines on the torus of revolution. Proposition 5.3. Consider a torus of revolution T(r, R)obtained by rotating a circle of radius rabout a line in the same plane and at a distance R,R>r, from its center. Define the function ρ=ρR r=1 2π2π 0 ds R r+ cos(s). Then the mean curvature lines on T(r, R)are all closed or all recurrent according to ρ∈Qor ρ∈R\Qand both cases occur. Proof: The torus of revolution T(r, R) is parametrized by α(s, θ)=((R+rcos(s)) cos(θ),(R+rcos(s)) sin(θ),rsin(s)). Direct calculation shows that E=r2,F=0,G=[R+rcos(s)]2and f= 0. Clearly (s, θ) is a principal chart. The differential equation of the mean curvature lines, in the principal chart (s, θ), is given by E(s, θ)ds2−G(s, θ)dθ2= 0. This is equivalent to r2ds2=[R+rcos(s)]2dθ2. Solving the equation above it is follows that, θ(2π)=θ0±2πρR r=θ0±2π 0 ds R r+ cos(s). Lines of Mean Curvature on Surfaces Immersed in R3447 So the two Poincar´e maps, π±:{s=0}→{s=2π}, defined by π±(θ0)=θ0±2πρ(R r) have rotation number equal to ±ρ. The function ρ(R r) is strictly decreasing and its image is the interval (0,∞), both the rational and irrational cases occur. This ends the proof. Remark 5.1.All the principal curvature lines are closed in the torus of revolution T(r, R). 5.3. Mean curvature lines on the ellipsoid. Proposition 5.4. Consider an ellipsoid Ea,b,c with three axes a>b> c>0. Then Ea,b,c have four umbilic points located in the plane of symmetry orthogonal to middle axis; they are of the type H1for mean curvature lines and of type D1for the principal curvature lines. Proof: Without lost of generality suppose that Ea,b,c is defined by the equation x2 a2+y2 b2+z2 c2= 1, with c= 1, and write A=1 a2and B=1 b2. Consider the parametrization of the ellipsoid α(x, y)=(x, y, h(x, y)) = (x, y, 1−Ax2−By2). Calculation shows that: E=1+Ax h2 F=ABxy h2G=1+By h2 e=−Ah2−A2x2 h3f=−ABxy hg=−Bh2−B2y2 h3. Therefore it is obtained that: L(x, y)=(Fg−Gf)(x, y)=AB(1 −B)xy h3 M(x, y)=(Eg −Ge)(x, y)=(A−B)+AB(1 −A)x2+AB(B−1)y2 h3 N(x, y)=(Ef −Fe)(x, y)=AB(A−1)xy h3. As A<B<1 it follows that the four umbilic points are: (±x0,0,±z0)=±B−A AB(1 −A),0,±A(1 −B) B(1 −A). 448 R. Garcia, J. Sotomayor In a neighborhood of the umbilic point (x0,0,z 0) it follows that the first order jet of the differential equation Ldy2+Mdxdy +Ndx2=0of principal curvature lines is given by: AB(1 −B)x0 z3 0 ydy2+2AB(1 −A)x0 z3 0 (x−x0)dxdy +AB(A−1)x0 z3 0 ydx2=0. Performing a change of coordinates x=x0+¯x,y=1−A 1−B¯ythe following equation is obtained: (1 −A)1−A 1−B[¯y(d¯y)2+2¯xd¯xd¯y−¯y(d¯x)2]=0. Therefore by Proposition 3.1 this umbilic point is of type D1and the same holds for the all the other umbilic points. In a neighborhood of the umbilic point (x0,0,z 0) the first order jet of the differential equation of the mean curvature lines is given by: [(1 −A)(x−x0)]dy2+ [2(A−1)y]dxdy +−(A−1)2 1−B(x−x0)dx2=0. In the differential equation above perform the change of coordinates x=x0+¯x,y=1−A 1−B¯yto obtain: (1 −A)2 1−B[¯x(d¯y)2−2¯yd¯xd¯y−¯x(d¯x)2]=0. Therefore by Proposition 3.2 the umbilic point is of type H1for the mean curvature lines. Proposition 5.5. Consider an ellipsoid Ea,b,c with three axes a>b> c>0. On the ellipse Σ⊂Ea,b,c, containing the four umbilic points, pi,i=1,...,4,counterclockwise oriented, denote by s1(resp. s2) the elliptic distance between the adjacent umbilic points p1and p4(resp. p1 and p2). Define ρ=s2 s1. Then if ρ∈R\Q(resp. ρ∈Q) all the mean curvature lines are recurrent (resp. all, with the exception of the mean curvature umbilic separatrices, are closed). See Figure 5.1. Lines of Mean Curvature on Surfaces Immersed in R3449 p1 p2 p3p4 p3 p2 p4 p1 Figure 5.1. Mean curvature lines on the ellipsoid. Proof: The ellipsoid Ea,b,c belongs to the triple orthogonal system of surfaces defined by the one parameter family of quadrics, x2 a2+λ+y2 b2+λ+ z2 c2+λ= 1 with a>b>c>0, see also [St] and [Sp]. The following parametrization of Ea,b,c α(u, v)=±a2(u+a2)(v+a2) (b2−a2)(c2−a2),±b2(u+b2)(v+b2) (b2−a2)(b2−c2), ±c2(u+c2)(v+c2) (c2−a2)(c2−b2) defines the ellipsoidal coordinates (u, v)onEa,b,c, where u∈(−b2,−c2) and v∈(−a2,−b2). The first fundamental form of Ea,b,c is given by: ds2=Edu2+Gdv2=1 4 (u−v)u (u+a2)(u+b2)(u+c2)du2 +1 4 (v−u)v (v+a2)(v+b2)(v+c2)dv2. The four umbilic points are (±x0,0,±z0)=(±aa2−b2 a2−c2,0,±cc2−b2 c2−a2). 450 R. Garcia, J. Sotomayor On the ellipse Σ = {(x, 0,z)|{ x2 a2+z2 c2=1}the distance between the umbilic points p1=(x0,0,z 0) and p4=(x0,0,−z0) is given by s1= −c2 −b2 √u (u+a2)(u+c2)du and that between the umbilic points p1=(x0,0,z 0) and p2=(−x0,0,z 0) is given by s2=−b2 −a2 √v (v+a2)(v+c2)dv. It is obvious that the ellipse Σ is the union of four umbilic points and four principal umbilical separatrices for the principal foliations. So Σ\{p1,p 2,p 3,p 4}is a transversal section of both mean curvature foliations. The differential equation of the mean curvature lines in the principal chart (u, v) is given by Edu2−Gdv2= 0, which is equivalent to (√Edu)2=( √Gdv)2, which amounts to ds1=±ds2. Therefore near the umbilic point p1the mean curvature lines with a mean curvature umbilic separatrix contained in the region {y>0}define a the return map σ+:Σ→Σ which is an isometry, reverting the orientation, with σ+(p1)=p1. This follows because in the principal chart (u, v) this return map is defined by σ+:{u=−b2}→{v=−b2}which satisfies the differential equation ds2 ds1=−1. By analytic continuation it results that σ+is a isometry reverting orientation with two fixed points {p1,p 3}. The geometric reflection σ−, defined in the region y<0 have the two umbilic {p2,p 4}as fixed points. So the Poincar´e return map π1:Σ→Σ (composition of two isometries σ+and σ−) is a rotation with rotation number given by s2 s1. Analogously for the other mean curvature configuration, with the Poincar´e return map given by π2=τ+◦τ−where τ+and τ−are two isometries having respectively {p2,p 4}and {p1,p 3}as fixed points. 6. Structural stability of mean curvature configurations Let M2be a compact, smooth and oriented surface. Denote by Mk,s be the space of Ckimmersions of Minto the Euclidean space R3, endowed with the Cstopology. Consider the subset Qkof immersions αdefined by the following conditions: a) all umbilic points are of types H1,H2or H3for mean curvature lines; b) all mean curvature cycles are hyperbolic; c) the limit set of every mean curvature line is contained in the set of umbilic points and mean curvature cycles of α; d) all mean curvature umbilic separatrices are associated to a single umbilic point; this means that there are no connections or self Lines of Mean Curvature on Surfaces Immersed in R3451 connections of mean curvature umbilic separatrices of both mean curvature configurations. Theorem 6.1. Let k≥4. The following holds: i) The subset Qkis open in Mk,3. ii) Every α∈Q kis Mean Curvature Structurally Stable. Proof: The openness of Qkit follows from the local structure of the mean curvature lines near the umbilic points Hi,i=1,2,3, near the mean curvature cycles and by the absence of umbilic mean curvature separatrix connections and the absence of recurrences. The equivalence can be performed by the method of canonical regions and their continuation as was done in [S-G], [GS2], for principal lines, and [Ga-Gu-S], for asymptotic lines. 7. Density of mean curvature structurally stable immersions In this section will be proved an approximation theorem for the class of immersions or surfaces having structurally stable mean curvature configuration. Theorem 7.1. Let k≥4. The subset Qkis dense in Mk,2. The proof of this theorem follows from the elimination of mean curvature recurrences and the stabilization of the mean curvature umbilical separatrices. The steps are basically those followed by C. Guti´errez and J. Sotomayor in the case of principal curvature lines, see [GS1], [GS2]. The main ideas goes back to M. Peixoto [Pe] and C. Pugh [Pu] to solve the similar problem of elimination of recurrences for vector fields on surfaces. See also the book by J. Palis and W. de Melo [P-M]. It will be established in what follows the main ingredients of the proof of the Approximation Theorem, with the complete proofs of the preliminary technical lemmas necessary to obtain the Lifting Lemma, essential to control the mean curvature lines under suitable deformation of the immersion. There is lost of generality to assume that the immersion is C∞or Cωin the proof of the density theorem. In what follows a chart whose coordinates lines are mean curvature lines will be called mean curvature chart for α. 452 R. Garcia, J. Sotomayor Lemma 7.1. Let α:M→R3be an immersion of class C∞and (u, v): (U, D)→(V,I ×I)be a positive mean curvature chart on M, where I=[−1,1]. Suppose that, for 9small, β=α=α+9ϕN is an immersion and ϕbe a smooth function on Uwhich satisfies: ϕ(−1,v)= ϕ(1,v)=ϕu(−1,v)=ϕu(1,v)=ϕuu(−1,v)=ϕuu(1,v)=0. Then the mean curvature line of αon Dwhich passes through qin {u= −1}∩{−1<v<1}meets the segment of abscissa {u=1}at a point whose v-coordinate vhas a derivative with respect to 9given by: (7.1) d d9(v)|=0 =1 −1 E 4√EG√H2−Kϕvvdu −1 −1 (EG)v 8fG2ϕvdu +1 −1f G−1 4fuu −(ln(√EG)u 4fuϕdu. Proof: Suppose that for 9small, β(u, v, 9)=α(u, v)=α(u, v)+9ϕ(u, v)N(u, v) is an immersion. The v-coordinate, v=v(u, q, 9), of the point where the line of mean curvature through the point qin {u=−1}∩{−1<v<1}meets the curve with abscissa {u}, satisfies the following Cauchy Problem with parameter 9. (7.2) [e(EG −2F2)+2EFf −E2g]+[4fEG−2EFg −2FGe]dv du +[g(EG −2F2)+2fFG−eG2]dv du2 =0,v(−1,9)=q. Since (u, v)isamean curvature chart it results that dv du(u, q, 0)=0, F(u, v, 0)=0, H(u, v, 0) = e E(u, v, 0) = g G(u, v, 0). (7.3) Lines of Mean Curvature on Surfaces Immersed in R3453 Differentiating the equation (7.2) with respect to 9, evaluated on (u,v(q),9), making 9=0 and using (7.3) it follows that dv d9 =∂v ∂9(u,q,9)|=0 satisfies the following Cauchy Problem: d dudv d9 =[e(EG)+e(EG)+2fEF−gE2+2gEE] [4fEG](u, v(q),0) dv d9 (−1,q,0)=0. (7.4) The structure equations for the immersion αare given by: Nu=fF −eG EG −F2αu+eF −fE EG −F2αv Nv=gF −fG EG −F2αu+fF −gE EG −F2αv αuu =Γ 1 11αu+Γ 2 11αv+eN αuv =Γ 1 12αu+Γ 2 12αv+fN αvv =Γ 1 22αu+Γ 2 22αv+gN. (7.5) The functions Γk ij are the Christoffel symbols whose expression in terms of Eand Gin a mean curvature chart are given by: Γ1 11 =Eu 2E,Γ2 11 =−Ev 2G,Γ1 12 =Ev 2E Γ2 12 =Gu 2G,Γ1 22 =−Gu 2E,Γ2 22 =Gv 2G. (7.6) By direct calculation, it is obtained βu=1−9ϕ e Eαu−9ϕ f Gαv+9ϕuN βv=−9ϕ f Eαu+1−9ϕ g Gαv+9ϕvN (7.7) 454 R. Garcia, J. Sotomayor βuu =−9(ϕH)u+Γ 1 11(1 −9ϕH)−9ϕ f GΓ1 12 −9ϕuHαu +(1 −9ϕH)Γ2 11 −9ϕf Gu−9ϕ f GΓ2 12 −9ϕu f Gαv (7.8) +(1 −9ϕH)e−9ϕf2 G+9ϕuuN βuv =−9(ϕH)v+(1−9ϕH)Γ1 12 −9ϕ f GΓ1 22 −9ϕu f Eαu +(1 −9ϕH)Γ2 12 −9ϕf Gv−9ϕf GΓ2 22 −9ϕuHαv (7.9) + [(1 −29ϕH)f+9ϕuv]N βvv =−9ϕf Ev−9ϕ f EΓ1 12 +(1−9ϕH)Γ1 22 −9ϕv f Eαu +−9ϕ f EΓ2 12 −9(ϕH)v+(1−9ϕH)Γ2 22 −9ϕvHαv (7.10) +−9ϕf2 E+(1−9ϕH)g+9ϕvvN. Also, ∂ ∂9(βu∧βv)|=0 =−2ϕHαu∧αv+ϕuN∧αv+ϕvαu∧N.(7.11) Therefore, using the equations (7.7)–(7.10) the following is obtained. E=−2ϕe, F=−2ϕf, G=−2ϕg e=ϕuu −3ϕeH−ϕuΓ1 11 −ϕvΓ2 11 −ϕf2 GN,αu∧αv g=ϕvv −3ϕHg−ϕf2 E−ϕuΓ1 22 −ϕvΓ2 22N,αu∧αv. (7.12) Let N=e(EG−2F2)+2fEF−gE2and M=4fEG−2gEF −2eFG. Then, using (7.6) and (7.12), it follows that N M|=0 =ϕuu 4f−ϕvvE 4fG −ϕu(EG)u 8fEG +(EG)vϕv 8fG2−fϕ G.(7.13) Lines of Mean Curvature on Surfaces Immersed in R3455 Using (7.13) when integrating the variational equation (7.4) and performing the partial integration with boundary conditions on the function ϕ, is achieved the expression for (dv d )|=0 as stated in (7.1). Lemma 7.2. Let α:M→R3be an immersion of class C∞and (u, v): (U, D)→(V,I ×I)be a positive mean curvature chart on M, where I=[−1,1]. Then there exists a smooth function ϕ:M→[0,1] whose support is contained in Dsuch that, if 9is small enough then, for every 9in [−r, r],β=α+9ϕN is an immersion and the mean curvature line for βon Dwhich passes through qin {u=−1}∩{−1<v<1} meets the segment {u=1}×{−1<v<1}at a point v(q)so that the map 9→v(q)is strictly increasing. Proof: Let mbe a real smooth function with values in [0,1], identically equal to 1 on a neighborhood of 0 and with support contained in I. Let ϕ=ϕ(u, v)=bv2 2m(u)m(v) and take r>0 small so that for any 9in [−r, r], β=α=α+9ϕN is a smooth immersion. Let v(u), u∈I, be the v-coordinate of the mean curvature lines of α=α+9ϕN such that v(q)=q. As ϕ(u, 0) = ϕv(u, 0) = 0 and ϕvv(u, 0) = bm(u)by Lemma 7.1 applied to the family of immersions αit follows that ∂v ∂9(u, 9)|(0,0) =1 −1 E 4√EG√H2−Km(u)du =c>0. This implies that the map 9→v(q) is strictly increasing. This proves the lemma. Lemma 7.3. Let α:M→R3be an immersion and (u, v): (U, D)→ (V,I ×I)be a positive mean curvature chart for αon M, where I= [−1,1]. Then given any η>0, there are numbers d, c ∈(0,1 12 )such that for every r∈(0,d]and qin {u=−1}∩{−1 2<v<1 2}, there exists a smooth function ϕ:M→[0,1] whose support is contained in Dr=v−1(v(q)+rI)and ||ϕ||2,V , the C2-norm of ϕon V, in the (u, v)-coordinate chart, is less than η. Furthermore, for every 9∈I,α=α+9ϕN is an immersion and the mean curvature line for αon Dwhich passes through qin {u= −1}∩{−1<v<1}meets the segment {u=1}∩{−1<v<1}at a point v(q)so that the map 9→v(q)is strictly increasing and its image contains the interval [v(q)−2c9, v(q)+2c9]. 462 R. Garcia, J. Sotomayor To eliminate the recurrent maximal mean curvature lines of Qα,2,it is necessary to perform the same deformation analysis as above, applied to this case, with no fundamental change. The mean curvature umbilical separatrices of Qα,1are stabilized taking care to consider the C2-deformations of the immersion α, with support in mean curvature charts disjoint from the nowhere dense set A (see Proposition 7.2) consisting of the minimal mean curvature umbilical separatrices and the minimal mean curvature cycles. So the stabilized minimal mean curvature umbilical separatrices and minimal mean curvature cycles are preserved and these deformations do not produce any new non trivial recurrence for the minimal mean curvature lines. Therefore the immersion αcan be approximated in the C2-topology by an immersion α1having all minimal and maximal mean curvature umbilical separatrices stabilized. a) b) D {s=x1} {s=x 2} {s=x 3} {s=x 1} {s=x2} {s=x3} D D z1 b1 a1 w1 x1 z 1 b 1 a 1 w 1 x 1x2 z2 b2 a2 w2 zn bn z n b n an wn a n w n xnx n Figure 7.2. Lifting of mean curvature lines. Part 2: Conclusion of the Proof of Theorem 7.1: The first step is to approximate an immersion αof compact and oriented surface Mby an immersion having all umbilic points of the type Hi, i=1,2,3. This can be done by the Transversality Theorem establishing the condition T=b(b−a)= 0 and by a finite number of small Lines of Mean Curvature on Surfaces Immersed in R3463 local changes on the coefficients of the third jet of the immersion at the umbilic points. Next approximate the immersion αin Cs-sense by an analytic immersion, which will be denoted by α1. There are two cases to consider. Case 1: The surface Mis diffeomorphic to a torus and the immersion α1 is without umbilics. By using Proposition 7.2 it is possible to obtain an analytic immersion α,C2close to α1having only finitely many mean curvature cycles, all of which have finite multiplicity. The resulting immersion αcan be deformed around a mean curvature cycle to a obtain an immersion with a hyperbolic mean curvature cycle. If this immersion is approximated by an analytic one, α, will have only finitely many mean curvature cycles, all of which with finite multiplicity. In either case, using Proposition 4.2, α can be approximated by an immersion ˜α, all whose mean curvature cycles are hyperbolic, which belong to the class Qk, since conditions i), iii) and iv) are guaranteed by the Stabilization Lemma 7.4. This ends the proof in this case. Case 2: The analytic immersion α1have umbilic points and all are of the types Hi,i=1,2,3. In this case, as shown in Part 1, the immersion α1can be taken so that both, minimal and maximal mean curvature umbilical separatrices are stabilized and without non trivial recurrences. The next step, using Proposition 4.2, is to deform the immersion in order to obtain an immersion with all minimal and maximal mean curvature cycles hyperbolic. This ends the proof. 8. Further developments and concluding remarks Let α:M→R3be an immersion of class Ck,k≥4. The principal configuration of αis the triple Pα={Uα,Pα,1,Pα,2}, where Pα,i,i=1,2, are respectively the minimal and maximal principal foliations of α. Denote by Pkthe class of immersions which are principal structurally stable. The main result on this subject was established in [S-G], [GS1], [GS2], where the following theorem was proved. Theorem 8.1 ([S-G], [GS1], [GS2]).For k≥4the subset Pkis open in Mk,3and dense in Mk,2. Therefore from Theorems 6.1, 7.1 and 8.1 the following will be established. 464 R. Garcia, J. Sotomayor Theorem 8.2. The subset Qk∩Pkis open in Mk,3and dense in Mk,2 for k≥4. Proof: The openness is clear. To prove the density take an immersion α∈P k. Assume that the umbilic points of αare simultaneously of types Diand Hi,i=1,2,3, see Proposition 3.3. Approximate αin the C2topology by an immersion in Qk, taking care so that the perturbations be localized away from the umbilic points, principal cycles and principal umbilic separatrices as in Section 7. Remark 8.1.Theorem 8.2 implies an important complement to a previous paper of the authors concerning axial configurations of immersions into R4. See the Introduction above and [Ga-So1]. Let α:T2→R3be an immersion of class Ck,k≥4, of the torus T2such that α∈Pk∩Qk and Uα=∅. Then αsatisfies the conditions for principal and mean axial structurally stability given by Theorem 6.1 in [Ga-So1]. Therefore the class of immersions of the torus in R4which are principal and mean axial stable is open and not empty in Mk. Remark 8.2.The main point of the forthcoming paper, [Ga-So2], is to modify the analysis above for the torus to the case of immersions α∈ Pk∩Q kwith Uα=∅to obtain immersions in R4which are principal and mean axial stable. References [B-F] J. W. Bruce and D. L. Fidal, On binary differential equations and umbilics, Proc. Roy. Soc. Edinburgh Sect. A 111(1-2) (1989), 147–168. [Da] G. Darboux,“Le¸cons sur la th´eorie g´en´erale des surfaces. Vol. IV: Sur la forme des lignes de courbure dans la voisinage d’un ombilic”. Note 07, Gauthier Villars, Paris, 1896. [Ga-Gu-S] R. Garcia, C. Guti´ errez and J. Sotomayor, Structural stability of asymptotic lines on surfaces immersed in R3,Bull. Sci. Math. 123(8) (1999), 599–622. [Ga-M-F-R] R. Garcia, D. K. H. Mochida, M. D. C. Romero Fuster and M. A. S. Ruas, Inflection points and topology of surfaces in 4-space, Trans. Amer. Math. Soc. 352(7) (2000), 3029–3043. [Ga-So1] R. Garcia and J. Sotomayor, Lines of axial curvature on surfaces immersed in R4,Differential Geom. Appl. 12(3) (2000), 253–269. Lines of Mean Curvature on Surfaces Immersed in R3465 [Ga-So2] R. Garcia and J. Sotomayor, A class of axial structurally stable immersions of surfaces into R4, in preparation. [G-G-T-G] C. Guti´ errez, I. Guadalupe, R. Tribuzy and V. Gu´ ı- ˜ nez, Lines of curvature on surfaces immersed in R4,Bol. Soc. Brasil. Mat. (N.S.) 28(2) (1997), 233–251. [GS1] C. Guti´ errez and J. Sotomayor, An approximation theorem for immersions with stable configurations of lines of principal curvature, in “Geometric dynamics” (Rio de Janeiro, 1981), Lecture Notes in Math. 1007, Springer, Berlin, 1983, pp. 332–368. [GS2] C. Guti´ errez and J. Sotomayor, Lines of curvature and umbilical points, in “Structurally Stable Configurations of Lines of Curvature and Umbilic Points on Surfaces”, Monografias del IMCA, Lima, Peru, 1998, IMPA, Rio de Janeiro, Brazil, 1991, pp. 1–84. [GS3] C. Guti´ errez and J. Sotomayor, Lines of curvature, umbilic points and Carath´eodory conjecture, Resenhas 3(3) (1998), 291–322. [Gu] V. Gu´ ı˜ nez, Positive quadratic differential forms and foliations with singularities on surfaces, Trans. Amer. Math. Soc. 309(2) (1988), 477–502. [Lit] J. A. Little, On singularities of submanifolds of higher dimensional Euclidean spaces, Ann. Mat. Pura Appl. (4) 83 (1969), 261–335. [P-M] J. Palis and W. de Melo,“Geometric theory of dynamical systems”, Springer-Verlag, New York, 1982. [Pe] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1(1962), 101–120. [Pu] C. C. Pugh, The closing lemma, Amer. J. Math. 89 (1967), 956–1009. [S-G] J. Sotomayor and C. Guti´ errez, Structurally stable configurations of lines of principal curvature, in “Bifurcation, Ergodic theory and applications” (Dijon, 1981), Ast´erisque 98–99, Soc. Math. France, Paris, 1982, pp. 195– 215. [Sp] M. Spivak,“A comprehensive introduction to differential geometry. Vol. III”, second ed., Publish or Perish Inc., Wilmington, Del., 1979. 466 R. Garcia, J. Sotomayor [SX1] B. Smyth and F. Xavier, A sharp geometric estimate for the index of an umbilic on a smooth surface, Bull. London Math. Soc. 24(2) (1992), 176–180. [SX2] B. Smyth and F. Xavier, Real solvability of the equation ∂2 zω=ρg and the topology of isolated umbilics, J. Geom. Anal. 8(4) (1998), 655–671. [St] D. J. Struik,“Lectures on classical differential geometry”, second ed., Dover Publications Inc., New York, 1988. Ronaldo Garcia: Instituto de Matem´atica e Estat´ıstica Universidade Federal de Goi´as C. Postal 131, Goiˆania, GO CEP 74001-970 Brazil E-mail address:[email protected] Jorge Sotomayor: Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo C. Postal 66281, S˜ao Paulo, SP CEP 05315-970 Brazil E-mail address:[email protected] Primera versi´o rebuda el 24 d’octubre de 2000, darrera versi´o rebuda el 9 de mar¸c de 2001.