Surfaces with zero mean curvature vector in 4-dimensional space forms

Transcription

1 Surfaces with zero mean curvature vector in 4-dimensional space forms Naoya Ando Abstract Equations for minimal surfaces in 4-dimensional Riemannian space forms and space-like surfaces in 4-dimensional Lorentz space forms with zero mean curvature vector were given in [22] and [1], respectively. They are written in terms of the curvatures of the normal connections of the surfaces. In the present paper, we will obtain pairs of elliptic partial differential equations of second order for the above-mentioned surfaces, in terms of the induced metrics and principal distributions with respect to suitably chosen normal vector fields. 1 Introduction Based on [18], [21, Chapter 7, Part C] and [14], Tribuzy-Guadalupe studied minimal surfaces in 4-dimensional Riemannian space forms, and gave equations for the surfaces in terms of the curvatures of the normal connections, according to whether principal curvatures at each point are dependent on or independent of the choice of a unit normal vector, in other words, whether the ellipse of curvature is everywhere or nowhere given by a circle ([22]). Alías-Palmer gave equations for space-like surfaces in 4-dimensional Lorentz space forms with zero mean curvature vector, in terms of the curvatures of the normal connections ([1]). The purpose of the present paper is to obtain pairs of elliptic partial differential equations of second order for the above-mentioned surfaces, in terms of the induced metrics and principal distributions with respect to suitably chosen normal vector fields. On a surface in a 3-dimensional Riemannian space form with no umbilical points and nowhere zero principal curvatures, the equations of Gauss and Codazzi yield an overdetermined system in the form of F u = α + βe F, F v = γ + δe F. The set of solutions of this system is determined by the induced metric and principal distributions on the surface and a principal curvature function gives a solution of the system. If there exists another solution, then it corresponds to a surface in the space which is isometric to but not congruent with the original surface so that principal directions at any point correspond to principal directions. Suppose that the system satisfies the compatibility condition. Then the set of solutions is given by a one-parameter family of functions. If the space is E 3, then the surface corresponding to each solution is said to be molding and has a family of lines of curvature which consists of geodesics ([15, pp ], [13, pp ], [7]). A 1

2 molding surface is parallel curved, that is, for each molding surface S, there exists a plane P in E 3 such that at each point of S, at least one principal direction is parallel to P ([3]). If the space is not Euclidean, then the compatibility condition does not necessarily imply the existence of a family of geodesics of curvature (see [8]). Suppose that the system does not satisfy the compatibility condition. Then the set of solutions consists of at most two functions. We can characterize a surface such that the system has just two solutions ([6], [8]) and we see that on a surface with nonzero constant mean curvature, if the system does not satisfy the compatibility condition, then the system has just two solutions. We can characterize a surface such that the system has a unique and multiple solution ([4], [8]) and we see that on a minimal surface, if the system does not satisfy the compatibility condition, then the system has a unique and multiple solution. If a 4-dimensional Riemannian space form N 4 is connected, simply connected and complete, then it is the Euclidean space, a sphere or a hyperbolic space. We denote by L 0 the constant sectional curvature of N 4. We will obtain the following theorem for a minimal surface in N 4 such that principal curvatures at each point depend on the choice of a unit normal vector. Theorem 1.1 Let (M,g) be a two-dimensional Riemannian manifold and suppose that the curvature K of (M,g) is less than a number L 0.LetD 1, D 2 be two one-dimensional distributions on M which are orthogonal to each other at any point of M. Then the following (a) and (b) are equivalent: (a) there exists an isometric minimal immersion of a connected neighborhood of each point of M into N 4 satisfying principal curvatures at each point depend on the choice of a unit normal vector, D 1, D 2 give principal directions corresponding to the maximum of the absolute values of principal curvatures at any point; (b) there exist isothermal coordinates (u, v) and functions p, q on a neighborhood of each point of M satisfying g = e 2p (du 2 + dv 2 ), / u D 1, / v D 2, and p uu + p vv = 1 e 2p cosh 2q L 0e 2p (1.1) q uu + q vv = 1 sinh 2q. (1.2) e2p In addition, if (a) and (b) hold, then the immersion which appears in (a) is uniquely determined by the metric g and distributions D 1, D 2 up to an isometry of N 4, and principal curvatures corresponding to D 1, D 2 are given by ±(1/e 2p )cosh q. 2

3 Remark There exists a local solution (p, q) of the system of equations (1.1) and (1.2) such that p, q are of only one variable u, and such a solution is uniquely determined by an initial data. Remark Any solution p of p uu + p vv =1/e 2p L 0 e 2p and q 0 satisfy (1.1) and (1.2). We will see that the surface corresponding to a pair of these p, q is contained in a totally geodesic hypersurface of the space N. Remark In the case where L 0 = 0, for a solution (p, q) of the system of equations (1.1) and (1.2), both p + q and p q are solutions of f uu + f vv =1/e 2f. Remark Suppose (a) in Theorem 1.1. We will see that the minimum of the absolute values of principal curvatures is given by (1/e 2p ) sinh q. Therefore setting w := u + 1v for isothermal coordinates (u, v) in (b) in Theorem 1.1, we see by cosh 2 q sinh 2 q = 1 that a holomorphic quartic differential Q defined by the immersion (see [9]) is represented as dw dw dw dw up to a nonzero constant. A complex curve in C 2 = E 4 is an isotropic minimal surface, i.e., a minimal surface with a property that principal curvatures at each point do not depend on the choice of a unit normal vector, and a connected isotropic minimal surface in E 4 is congruent with a complex curve in C 2 = E 4. The isotropicity of a minimal surface in N 4 is equivalent to a condition that a holomorphic quartic differential Q defined on the surface vanishes ([9]). In the case where N 4 = S 4, this differential coincides with a holomorphic quartic differential given in [11] and therefore a superminimal surface is just an isotropic minimal surface. We will obtain the following theorem for an isotropic minimal surface in N 4. Theorem 1.2 Let (M,g) be a two-dimensional Riemannian manifold and suppose that the curvature K of (M,g) is less than a number L 0. Then the following (a) and (b) are equivalent: (a) there exists an isometric isotropic minimal immersion of a connected neighborhood of each point of M into N 4 ; (b) for isothermal coordinates (u, v) on a neighborhood of each point of M, there exist functions p, q satisfying g = e 2p (du 2 + dv 2 ), and p uu + p vv =2 e2q e 2p L 0e 2p (1.3) q uu + q vv = 2 e2q. (1.4) e2p In addition, if (a) and (b) hold, then the immersion which appears in (a) is uniquely determined by the metric g up to an isometry of N 4, and principal curvatures at each point are given by ±e q /e 2p. 3

4 Remark In the proof of Theorem 1.2, we will see that if (a) holds, then for given isothermal coordinates (u, v), we can choose a unit normal vector field ν such that / u and / v give principal distributions with respect to ν. Remark There exists a local solution (p, q) of the system of equations (1.3) and (1.4) such that p, q are of only one variable u, and such a solution is uniquely determined by an initial data. Remark Let p be a solution of p uu + p vv = (L 0 /3)e 2p. If L 0 > 0, then p and q := 2p +(1/2) log(l 0 /3) satisfy (1.3) and (1.4), and the surface corresponding to a pair of these p, q satisfy K L 0 /3. We will see that the surface is part of the Veronese surface. Remark Suppose L 0 = 0. Then for a solution (p, q) of the system of equations (1.3) and (1.4), s := p + q satisfies s uu + s vv = 0, and t := p q satisfies t uu + t vv =4/e 2t. We see that a system of these equations can be considered as a rewrite of the characterization of a complex curve in C 2 in terms of the induced metric and a holomorphic cubic differential given in [5], on condition that the intrinsic curvature is negative. Remark Let N be an oriented 4-dimensional Riemannian manifold. Let h be the metric of N and the Levi-Civita connection of h. Let M be a connected Riemann surface and F : M N a conformal immersion of M into N. Let w = u + 1v be a local complex coordinate on a neighborhood of each point of M and set T 1 := df ( / u), T 2 := df ( / v). In the case where F is an isotropic minimal immersion, F is said to be compatible with the orientation of N if (T 1,T 2, T1 T 1, T1 T 2 ) is an ordered basis of the tangent space of N which gives the orientation of N for a point of M where principal curvatures of F are nonzero. Suppose that N is a Kähler surface. Then F is a holomorphic immersion if and only if F is a real-analytic isotropic minimal immersion with at least one complex point and compatible with the orientation of N ([10]). Suppose that N is a hyperkähler 4-manifold and that I, J, K := IJ are complex structures on N such that (h, I, J, K) is a hyperkähler structure of N. Then referring to [16] or from [10], we see that F is a holomorphic immersion with respect to a complex structure ai + bj + ck for an element (a, b, c) ofs 2 if and only if F is an isotropic minimal immersion compatible with the orientation of N. If a 4-dimensional Lorentz space form N1 4 is connected, simply connected and complete, then it is the Minkowski space, a de Sitter space or an anti-de Sitter space. See [20] for pseudo-riemannian manifolds. We denote by L 0 the constant sectional curvature of N1 4. Referring to [12], we see that a Willmore surface in S 3 with no umbilical points is just given by a space-like surface in the de Sitter 4-space S1 4 with zero mean curvature vector and nowhere zero principal curvature with respect to a light-like normal vector field. We will obtain the following theorem for a space-like surface in N1 4 with zero mean curvature vector. 4

5 Theorem 1.3 Let (M, g) be a two-dimensional Riemannian manifold and suppose that the curvature K of (M, g) is nowhere equal to a number L 0. Let D 1, D 2 be two onedimensional distributions on M which are orthogonal to each other at any point of M. Then the following (a) and (b) are equivalent: (a) there exists an isometric immersion of a connected neighborhood of each point of M into N1 4 with zero mean curvature vector such that D 1, D 2 give principal directions with respect to a light-like normal vector field ν at any point; (b) on a neighborhood of each point of M, there exist isothermal coordinates (u, v) and functions φ, θ satisfying g = e 2φ (du 2 + dv 2 ), cos(θ/2) / u sin(θ/2) / v D 1, and In addition, if (a) and (b) hold, then φ uu + φ vv = 2 e 2φ cos 2θ L 0e 2φ (1.5) θ uu + θ vv = 2 sin 2θ. (1.6) e2φ the immersion which appears in (a) is uniquely determined by the metric g and distributions D 1, D 2 up to an isometry of N1 4, we can suppose that θ satisfies θ ( π/4,π/4) or θ (π/4, 3π/4), if ι is a light-like normal vector field of the immersion in (a) satisfying ι, ν = 1, where, is the Lorentz metric of N1 4, then cos(θ/2) / u + sin(θ/2) / v gives a principal direction with respect to ι at any point, the product of a principal curvature with respect to ι and a principal curvature with respect to ν is given by ±1/e 4φ. Remark In [17], we can find a system of differential equations corresponding to the system of equations (1.5) and (1.6), in the case where the surface is time-like. Remark There exists a local solution (φ, θ) of the system of equations (1.5) and (1.6) such that φ, θ are of only one variable u, and such a solution is uniquely determined by an initial data. Remark For θ 0 (π/2)z, there exists a local solution φ of (1.5) with θ := θ 0. Then φ and θ θ 0 satisfy (1.6). We will see that the surface corresponding to a pair of these φ, θ is contained in a totally geodesic hypersurface of the space N1 4. 5

6 Remark In the case where L 0 = 0, the system of equations (1.5) and (1.6) is rewritten into ξ ww = e ξ, where ξ := 2(φ + 1θ). This is an analogue of Liouville s equation and the general solution of ξ ww = e ξ is given by e ξ = 2S (w)t (w) (S(w)+T (w)) 2, where S, T are holomorphic functions of w, w, respectively. Remark Let M be an oriented two-dimensional manifold and ι : M S 3 an immersion of M into S 3. The conformal Gauss map γ ι of ι is a smooth map from M into S1 4 and if we denote by Reg (ι) the set of non-umbilical points of ι, then γ ι Reg (ι) is a spacelike immersion and induces a Riemannian metric g on Reg (ι) which is conformal to the induced metric g by ι as follows: g =(H 2 K +1)g, where H is the mean curvature of ι, and K is the intrinsic curvature of (M,g). An immersion ι : M S 3 is Willmore if and only if the mean curvature vector of γ ι Reg (ι) is identically zero ([12]), and for a space-like immersion γ : M S1 4 with zero mean curvature vector and nowhere zero principal curvature with respect to a light-like normal vector field, there exists a Willmore immersion ι : M S 3 such that the conformal Gauss map γ ι is given by the above γ up to a sign. We can define analogues of the conformal Gauss map of a surface in S 3 : the conformal Gauss map of a surface in H 3 is a map into S1; 4 the conformal Gauss maps of space-like surfaces in the de Sitter and the anti-de Sitter 3-spaces S1 3, H3 1 are maps into the anti-de Sitter 4-space H1; 4 the conformal Gauss map of a space-like surface in the cone of future-directed light-like vectors of the Minkowski 4-space E1 4 is a map into E4 1 (see [9]). The image of a Willmore sphere in S 3 by a stereographic projection is just given by the compactification of the image of a connected, complete minimal surface in E 3 with finite total curvature and embedded, flat ends by a conformal transformation of E 3 ([12]). See [19] for the case where the ambient space is S 4. For a Willmore immersion ι : M S 3 with no umbilical points, the curvature of the induced metric by the conformal Gauss map γ ι is identically equal to one if and only if the composition of ι with a stereographic projection is a minimal immersion into E 3 ([2]). Remark Suppose (a) in Theorem 1.3. Then for light-like normal vector fields ι, ν of the immersion with ι, ν = 1, the product of principal curvatures with respect to ι, ν is given by ±1/e 4φ. Therefore noticing that cos(θ/2) / u + sin(θ/2) / v (respectively, cos(θ/2) / u sin(θ/2) / v) gives a principal direction with respect to ι (respectively, ν) at any point, we see that a holomorphic quartic differential Q defined by the immersion (see [9]) is represented as dw dw dw dw up to a nonzero constant. 6

7 2 Minimal surfaces in 4-dimensional Riemannian space forms 2.1 The equations of Gauss, Codazzi and Ricci Let N be a 4-dimensional Riemannian space form with constant sectional curvature L 0. Let M be an oriented two-dimensional manifold and ι : M N a minimal immersion of M into N. Let e 1, e 2 form an orthonormal frame field on a neighborhood U of a point of M and suppose that (e 1,e 2 ) gives the orientation of M. Let ν 1, ν 2 be unit normal vector fields of ι : U N which are perpendicular to each other at any point. If L 0 = 0, then we can consider N to be E 4 and therefore e 1, e 2, ν 1, ν 2 to be E 4 -valued functions on U which form a basis of T ι(a) (E 4 ) for any a U. If L 0 > 0, then we can consider N to be a hypersphere in E 5 with radius 1/ L 0 centered at the origin of E 5 ;ifl 0 < 0, then we can suppose N = {x E1 5 x, x =1/L 0}, where E1 5 is a connected, simply connected, complete and flat Lorentz space form: E1 5 is diffeomorphic to R 5 and equipped with the following indefinite metric, : for x := (x 0,x 1,x 2,x 3,x 4 ), y := (y 0,y 1,y 2,y 3,y 4 ) R 5, x, y := x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 + x 4 y 4. Therefore if L 0 0, then we can consider ι, e 1, e 2, ν 1, ν 2 to be R 5 -valued functions on U which form a basis of T ι(a) (R 5 ) for any a U. For θ R, we set ν(θ) := (cos θ)ν 1 + (sin θ)ν 2. Then ν(θ) is a unit normal vector field of ι. Let k(θ) be a principal curvature of ι with respect to ν(θ) and set k 1 := max{ k(θ) θ R} at each a U. We call k 1 the maximal principal curvature of ι at a. Let θ R satisfy k(θ) = k 1. Then an eigendirection of the shape operator A ν(θ) of ι with respect to ν(θ) is called a maximal principal direction of ι at a. In the following, we suppose k 1 > 0 and k(0) = k 1 on U. In addition, suppose that e 1, e 2 give maximal principal directions of ι at any point of U and suppose A ν1 (e 1 )=k 1 e 1 and A ν1 (e 2 )= k 1 e 2 on U. We set k 2 := k(π/2). Then exchanging ν 2 for ν 2 if necessary, we obtain A ν2 (e 1 )=k 2 e 2 and A ν2 (e 2 )=k 2 e 1 on U. Let ω0, 1 ω0 2 be 1-forms on U defined by dι = e 1 ω0 1 + e 2 ω0. 2 Then ω0, 1 ω0 2 form the dual frame field of (e 1,e 2 ). We obtain (dι de 1 de 2 dν 1 dν 2 ) 0 L 0 ω0 1 L 0 ω ω ω1 2 k 1 ω0 1 k 2 ω0 2 =(ιe 1 e 2 ν 1 ν 2 ) ω0 2 ω1 2 0 k 1 ω0 2 k 2 ω0 1 0 k 1 ω0 1 k 1 ω0 2 0 α 0 k 2 ω0 2 k 2 ω0 1 α 0, (2.1) 7

8 where ω1, 2 α are 1-forms on U. We can represent ω1 2 as ω1 2 = l 1 ω0 1 + l 2 ω0, 2 where l i is the geodesic curvature of each integral curve of e i. We set α := α 1 ω0 1 + α 2ω0 2. From ddι =0, we obtain By dde 1 = 0 together with (2.2), we obtain dω 1 0 = l 1ω 1 0 ω2 0, dω2 0 = l 2ω 1 0 ω2 0. (2.2) K = k 2 1 k2 2 + L 0, e 2 (k 1 )=2l 1 k 1 α 1 k 2, e 1 (k 2 )= 2l 2 k 2 + α 2 k 1, (2.3) where K is the curvature of (M,g) and therefore given by By dde 2 = 0 together with (2.2), we obtain Kω 1 0 ω2 0 = dω2 1. K = k 2 1 k L 0, e 1 (k 1 )= 2l 2 k 1 + α 2 k 2, e 2 (k 2 )=2l 1 k 2 α 1 k 1. (2.4) The first equation of (2.3) is just the first equation of (2.4) and the equation of Gauss. Since k 1 > 0, we obtain K<L 0 from the equation of Gauss (conversely, if K<L 0, then k 1 > 0). The others in (2.3) and (2.4) than the equation of Gauss are the equations of Codazzi and we can represent them as dk =2k ω k α, (2.5) where k := k 1 + 1k 2 and is Hodge s -operator. The last equation, which is obtained from ddν i =0,is e 1 (α 2 ) e 2 (α 1 )+l 1 α 1 + l 2 α 2 = 2k 1 k 2. (2.6) We see that (2.6) is just the equation of Ricci and represented as dα = 2k 1 k 2 ω 1 0 ω 2 0. (2.7) 2.2 Proof of Theorem 1.2 Suppose k 1 k 2 on U and set k 0 := k 1 = k 2, which is a function on U and not necessarily constant. Then we can represent the equations of Gauss, Codazzi and Ricci as K L 0 = 2k0 2, dlog k 0 =2 ω1 2 α, dα = 2k2 0 ω1 0 ω2 0. (2.8) Let (u, v) be isothermal coordinates on U. Then there exists a function θ on U such that ι uv is perpendicular to ν(θ) at any point of U. Therefore we can suppose that ι uv is perpendicular to ν 1 and that there exists a function p on U satisfying e 1 = e p / u, e 2 = e p / v. Then we obtain ω1 2 = dp and by this together with the second equation 8

9 of (2.8), we obtain α = d(log k 0 +2p). If we set q := 2p + log k 0, then from the first equation of (2.8), we obtain (1.3) and from the third equation of (2.8), we obtain (1.4). Let p, q be functions of two variables u, v on a domain D of R 2 satisfying (1.3) and (1.4). We set ω0 1 := e p du, ω0 2 := e p dv, ω1 2 := dp, α := dq, k 1 = k 2 := eq e. 2p Suppose L 0 = 0. Let Ω be a 4 4 matrix such that the components are 1-forms on D as follows: 0 ω1 2 k 1 ω0 1 k 2 ω0 2 ω1 2 0 k 1 ω0 2 k 2 ω 1 0 Ω:= k 1 ω0 1 k 1 ω0 2 0 α. k 2 ω0 2 k 2 ω0 1 α 0 Then Ω satisfies dω +Ω Ω = 0. This is the compatibility condition of a linear over-determined system dx = XΩ and therefore we have a unique local solution X = (X 1 X 2 X 3 X 4 )ofdx = XΩ for a given initial value, where each X i is an R 4 -valued function. Then Y := t XX is a solution of a linear over-determined system dy = t ΩY + Y Ω. The compatibility condition of dy = t ΩY + Y Ω is given by dω+ω Ω = 0. Suppose that Y is equal to the unit matrix I 4 of degree 4 at a point. Then Y identically coincides with I 4, because Y I 4 is a solution of dy = t ΩY + Y Ω. Since d(x 1 ω0 1 + X 2 ω0) 2 = 0, we can locally find an R 4 -valued function ι satisfying dι = X 1 ω0 1 +X 2ω0 2. In addition, ι, e 1 := X 1, e 2 := X 2, ν 1 := X 3, ν 2 := X 4 satisfy (2.1). Suppose L 0 0 and let Ω be a 5 5 matrix such that the components are 1-forms on D as in the right side of (2.1). Then Ω satisfies dω+ω Ω = 0. Therefore we have a unique local solution X =(X 1 X 2 X 3 X 4 X 5 )of dx = XΩ for a given initial value, where each X i is an R 5 -valued function. We set L 0 / L T := Then Y := t XTX is a solution of dy = t ΩY + Y Ω. Suppose that Y is equal to a diagonal matrix diag (1/L 0, 1, 1, 1, 1) at a point. Then Y identically coincides with diag (1/L 0, 1, 1, 1, 1). We see that ι := X 1, e 1 := X 2, e 2 := X 3, ν 1 := X 4, ν 2 := X 5 satisfy (2.1). Hence there exists a neighborhood U of each a D satisfying ι : U N is a minimal immersion, the induced metric g by ι is given by g = e 2p (du 2 + dv 2 ) and the curvature K of (U, g) is less than L 0, 9

10 ν 1, ν 2 are unit normal vector fields of ι : U N which are perpendicular to each other at any point, and principal curvatures of ι at each point of U do not depend on the choice of a unit normal vector and they are given by ±e q /e 2p. Therefore we obtain Theorem 1.2. Example The Veronese surface is a minimal surface in S 4 with positive constant intrinsic curvature 1/3, and it has the property that principal curvatures do not depend on the choice of a unit normal vector at any point. Let S 2 (1/3) be a two-dimensional sphere with constant curvature 1/3. Considering S 2 (1/3) to be the sphere in E 3 centered at the origin with radius 3, let π : S 2 (1/3) \{(0, 0, 3)} R 2 be the stereographic projection from (0, 0, 3). Then π 1 induces the metric of S 2 (1/3) on R 2 and it is represented as g = 12 (1 + u 2 + v 2 ) 2 (du2 + dv 2 ), where (u, v) are the standard coordinates on R 2. We know that π 1 : R 2 S 2 (1/3) is represented as 3 π 1 (u, v) = 1+u 2 + v (2u, 2v, 1 2 u2 v 2 ) ((u, v) R 2 ). Let F : E 3 E 5 be a map which assigns to each (x, y, z) E 3 a point of E 5 given by ( 1 3 yz, 1 3 zx, 1 3 xy, (x2 y 2 ), ) 1 6 (x2 + y 2 2z 2 ). Then the restriction F S 2 (1/3) of F on S 2 (1/3) is an isometric, minimal immersion into S 4 and the image is just the Veronese surface. For (u, v) R 2, 2 3v(1 u 2 v 2 ) 2 3u(1 u F π v 2 ) (u, v) = (1 + u 2 + v 2 ) 2 4 3uv 2. (2.9) 3(u 2 v 2 ) 1+4(u 2 + v 2 ) (u 2 + v 2 ) 2 We set ( ) ( ) e 1 := 1+u2 + v 2 2 d(f π 1 ), e 2 := 1+u2 + v 2 3 u 2 d(f π 1 ). 3 v Then e 1, e 2 form an orthonormal frame field. We set T i := (1 + u 2 + v 2 ) 2 e i for i =1, 2. Then by (2.9), we see that each component of T i is a polynomial of u, v, and for the Levi-Civita connection of E 5, we obtain T1 T 1, T1 T 1 T1 T 2, T1 T 2 = 12(1 + u 2 + v 2 ) 2, T1 T 1, T1 T 2 =0. 10

11 These imply ( e1 e 1 ), ( e1 e 1 ) = ( e1 e 2 ), ( e1 e 2 ), ( e1 e 1 ), ( e1 e 2 ) =0, respectively, where ( e1 e i ) is the normal component of e1 e i in S 4. Therefore principal curvatures of F S 2 (1/3) do not depend on the choice of a unit normal vector at any point and they are identically equal to ±1/ 3. Remark Let p be a solution of p uu +p vv = (1/3)e 2p. Then p and q := 2p+(1/2) log(1/3) satisfy (1.3) with L 0 = 1 and (1.4). For these p, q, we can construct a unique immersion as above up to an isometry of the space. Then the intrinsic curvature is identically equal to 1/3 and noticing the above example, we see that the surface is part of the Veronese surface. 2.3 Proof of Theorem 1.1 In the following, we suppose k 1 > k 2. Then by (2.5), we obtain 0=ddk =2dk ω1 2 +2kd ω2 1 1dk α 1 kd α =2kd ω1 2 1 kd α. (2.10) Since k1 2 k2 2 0, from (2.10), we obtain d ω2 1 = 0 and d α = 0. Therefore there exist functions p and q on U satisfying dp = ω1 2 and dq = α. From dp = ω1, 2 we see that there exist isothermal coordinates (u, v)onu satisfying e 1 = e p / u, e 2 = e p / v. Then from dq = α and (2.7), we obtain 1 e (q 2p uu + q vv )= 2k 1 k 2. (2.11) Noticing the first equation of (2.3), we find a function θ on U satisfying k 1 = L 0 K cos θ, k 2 = L 0 K sin θ. (2.12) Therefore by k = L 0 Ke 1θ together with (2.5), we obtain d log L 0 K = 2dp + sin 2θdq, dθ = cos 2θdq. (2.13) From the second equation of (2.13), we can suppose that θ and q satisfy sin 2θ = tanh 2q. (2.14) Therefore by the first equation of (2.13), we can suppose that p satisfies L 0 K = 1 cosh 2q. (2.15) e4p 11

12 From (2.15), we obtain (1.1). From (2.11), (2.12), (2.14) and (2.15), we obtain (1.2). Let p, q be functions of two variables u, v on a domain D of R 2 satisfying (1.1) and (1.2). We set ω 1 0 := e p du, ω 2 0 := e p dv, ω 2 1 := dp, α := dq and k 1 + ( ) cosh 2q 1 1k 2 := exp sin 1 (tanh 2q) e 2p 2 = 1 e (cosh q + 1sinh q). 2p Then referring to Subsection 2.2, we obtain ι, e 1, e 2, ν 1, ν 2 similarly and we see that there exists a neighborhood U of each a D satisfying ι : U N is a minimal immersion, the induced metric g by ι is given by g = e 2p (du 2 + dv 2 ) and the curvature K of (U, g) is less than L 0, ν 1, ν 2 are unit normal vector fields of ι : U N which are perpendicular to each other at any point, and principal curvatures of ι with respect to ν i at each point of U are given by ±k i and k 1 is the maximal principal curvature. Therefore we obtain Theorem 1.1. Remark If q 0, then α vanishes and either k 1 or k 2 is identically zero. Therefore noticing (2.1), we see that for a solution p of p uu + p vv =1/e 2p L 0 e 2p and q 0, the surface is contained in a totally geodesic hypersurface of N. Remark In Subsection 2.2 and Subsection 2.3, we discussed the cases k 1 k 2 and k 1 > k 2, respectively. On the other hand, it is possible that there exists a point satisfying k 1 = k 2 although k 1 k 2. 3 Space-like surfaces with zero mean curvature vector in 4-dimensional Lorentz space forms 3.1 The equations of Gauss, Codazzi and Ricci For a real number L 0, let N1 4 be a connected, simply connected and complete 4-dimensional Lorentz space form with constant sectional curvature L 0. If L 0 = 0, then N1 4 is the 12

13 Minkowski 4-space E1. 4 If L 0 > 0, then N1 4 is a de Sitter space, which we denote by S1 4(L 0) and we can represent S1 4(L 0)as { S1 4 (L 0)= x E1 5 x, x = 1 }, L 0 where E 5 1 is as in Subsection 2.1, and therefore we have S4 1 (1) = S4 1. The metric of S4 1 (L 0) is the restriction of the metric of E 5 1 on S4 1 (L 0). Let, be an indefinite metric of R 5 defined by x, y := x 0 y 0 x 1 y 1 + x 2 y 2 + x 3 y 3 + x 4 y 4 for x := (x 0,x 1,x 2,x 3,x 4 ), y := (y 0,y 1,y 2,y 3,y 4 ) R 5. Let E2 5 be a pseudo-riemannian manifold which is diffeomorphic to R 5 and equipped with the indefinite metric,. If L 0 < 0, then N1 4 is an anti-de Sitter space, which we denote by H1(L 4 0 ) and we can represent H1 4(L 0)as { H1(L 4 0 )= x E2 5 x, x = 1 }, L 0 and therefore we have H1(1) 4 = H1. 4 The metric of H1(L 4 0 ) is the restriction of the metric of E2 5 on H4 1 (L 0). We denote the metric of N1 4 by,. Let M be an oriented two-dimensional manifold. Let γ : M N1 4 be a space-like immersion of M into N1 4 with zero mean curvature vector. Then there exist light-like normal vector fields ι, ν of γ satisfying ι, ν = 1. In the following, we suppose that principal curvatures of γ with respect to ι or ν are nonzero. Then we can suppose that principal curvatures with respect to ι are identically equal to ±1. Let ẽ 1,ẽ 2 be vector fields on a neighborhood U of a point of M satisfying g(ẽ i, ẽ j )=δ ij, where g is the induced metric by γ, (ẽ 1, ẽ 2) gives the orientation of M, A ι (ẽ 1 )=ẽ 1, A ι(ẽ 2 )= ẽ 2, where A ι is the shape operator of γ with respect to ι. Let ẽ 1,ẽ 2 be vector fields on U satisfying g(ẽ i, ẽ j )=δ ij, (ẽ 1, ẽ 2 ) gives the orientation of M, A ν (ẽ 1 )=kẽ 1, A ν (ẽ 2 )= kẽ 2, where k is a function on U. Let θ be a function on U satisfying ẽ 1 = (cos θ)ẽ 1 + (sin θ)ẽ 2, ẽ 2 = (sin θ)ẽ 1 + (cos θ)ẽ 2. (3.1) Therefore from A ι (ẽ i)=( 1) i 1 ẽ i and (3.1), we obtain A ι (ẽ 1 ) = (cos 2θ)ẽ 1 + (sin 2θ)ẽ 2, A ι (ẽ 2 ) = (sin 2θ)ẽ 1 (cos 2θ)ẽ 2. (3.2) 13

14 If L 0 = 0, then we obtain N1 4 = E1 4 and therefore we can consider ẽ 1,ẽ 2, ι, ν to be E1 4-valued functions on U which form a basis of T γ(a)(e1 4) for any a U. If L 0 0, then we can consider γ, ẽ 1,ẽ 2, ι, ν to be R 5 -valued functions on U which form a basis of T γ(a) (R 5 ) for any a U. Therefore by (3.2), we obtain (dγ dẽ 1 dẽ 2 d ι d ν) 0 L 0 ω 0 1 L 0 ω ω ω 1 2 ω 2θ k ω 0 1 =(γ ẽ 1 ẽ 2 ι ν) ω 0 2 ω ω 2θ k ω k ω 0 1 k ω 0 2 β 0 0 ω 2θ ω 2θ 0 β, (3.3) where ω 0 1, ω2 0 form the dual frame field of (ẽ 1, ẽ 2 ), ω 1 2 is a 1-form on U represented as ω2 1 = l 1 ω 0 1+ l 2 ω 0 2, where l i is the geodesic curvature of each integral curve of ẽ i, ω 2θ := (cos 2θ) ω (sin 2θ) ω2 0, ω 2θ := (sin 2θ) ω1 0 (cos 2θ) ω2 0, β is a 1-form on U. From ddγ = 0, we obtain d ω 0 1 = l 1 ω 0 1 ω2 0, d ω2 0 = l 2 ω 0 1 ω2 0. (3.4) By (3.4), we obtain d ω 1 2 = K ω 0 1 ω2 0, d ω 2θ = (2dθ + ω 1 2 ) ω 2θ, d ω 2θ =(2dθ + ω2 1 ) ω 2θ, (3.5) where K is the curvature of (M, g). Using (3.4), the first equation of (3.5) and ddẽ i =0, we obtain the equation of Gauss: K =2k cos 2θ + L 0. (3.6) If k 0 and if θ (1/4+n/2)π for any n Z, then from (3.6), we obtain K L 0. If K L 0, then from (3.6), we see that principal curvatures with respect to any light-like normal vector field are nonzero and that θ (1/4+n/2)π for any n Z. Using (3.4), the second and the third equations of (3.5) and ddẽ i = 0, we obtain the equations of Codazzi: (ẽ 2 (k) 2k l 1 ) ω 1 0 ω k ω 1 0 β =0, (ẽ 1 (k)+2k l 2 ) ω 1 0 ω2 0 k ω2 0 β =0 (3.7) 14

15 and 2( ω dθ) ω 2θ ω 2θ β =0, 2( ω dθ) ω 2θ + ω 2θ β =0. (3.8) In the following, we suppose k>0and we set φ := (1/4) log k. Then we can rewrite (3.7) and (3.8) into β =2 ω dφ = 2 ω dθ. (3.9) By dd ι =0ordd ν = 0, we obtain the equation of Ricci: 3.2 Proof of Theorem 1.3 dβ = 2 e 4φ sin 2θ ω1 0 ω2 0. (3.10) Let (u, v) be isothermal coordinates on U which give the orientation of M and we represent g as g = λ 2 (du 2 + dv 2 ), where λ is a positive-valued function on U. From (3.9), we obtain dβ =2d ω 1 2 = d dθ. Therefore by (3.10), we obtain 1 λ (θ 2 uu + θ vv )= 2 sin 2θ. (3.11) e4φ From (3.9), we obtain ω 1 2 = dφ 1 dθ. (3.12) 2 Therefore by the first equation of (3.5), (3.6) and (3.12), we obtain 1 λ (φ 2 uu + φ vv )= 2 e cos 2θ L 0. (3.13) 4φ We set ê 1 := (1/λ) / u, ê 2 := (1/λ) / v. Let t be a function on U satisfying ẽ 1 = (cos t)ê 1 + (sin t)ê 2, ẽ 2 = (sin t)ê 1 + (cos t)ê 2. (3.14) Let ˆω 1 0,ˆω 2 0 form the dual frame field of (ê 1, ê 2 ). Then we obtain ω 1 0 = (cos t)ˆω1 0 + (sin t)ˆω2 0, ω2 0 = (sin t)ˆω1 0 + (cos t)ˆω2 0. Let be the Levi-Civita connection with respect to g. Then the vector field corresponding to ω 1 2 by g is given by l 2 ẽ 1 + l 1 ẽ 2 = ẽ1 ẽ 1 + ẽ2 ẽ 2 and by (3.14), we obtain ẽ1 ẽ 1 + ẽ2 ẽ 2 = ê1 ê 1 + ê2 ê 2 ê 2 (t)ê 1 +ê 1 (t)ê 2. (3.15) Therefore from (3.12) and (3.15), we obtain dt +(ˆl 1ˆω ˆl 2ˆω 0 2 )= dφ 1 dθ, (3.16) 2 15

16 where ˆl i is the geodesic curvature of each integral curve of ê i. Therefore we obtain 1 λ ((log λ) 2 uu + (log λ) vv ) ω 0 1 ω2 0 = K ω 0 1 ω2 0 = d dφ and therefore h := log λ φ satisfies h uu + h vv = 0. Therefore choosing isothermal coordinates (u, v), we can suppose λ = e φ. Then we can rewrite (3.13) and (3.11) into (1.5) and (1.6), respectively. From λ = e φ and (3.16), we obtain d(θ +2t) = 0. Therefore we can suppose t = θ/2. Let φ, θ be functions of two variables u, v on a domain D of R 2 satisfying (1.5) and (1.6). We set ω 1 0 := e φ (cos(θ/2)du sin(θ/2)dv), ω 2 0 := e φ (sin(θ/2)du + cos(θ/2)dv), ω 2 1 := dφ 1 2 dθ, k := e 4φ, β := 2dφ dθ. Suppose L 0 = 0 and let Ω be a 4 4 matrix such that the components are 1-forms on D as follows: 0 ω 1 2 ω 2θ k ω 0 1 ω ω 2θ k ω 2 0 Ω:= k ω 0 1 k ω 0 2 β 0. ω 2θ ω 2θ 0 β Then Ω satisfies dω + Ω Ω = 0. Therefore we have a unique local solution X = (X 1 X 2 X 3 X 4 )ofdx = XΩ for a given initial value, where each X i is an R 4 -valued function. We set T 0 := Then Y := t XT 0 X satisfies dy = t ΩY + Y Ω. Suppose that Y is equal to Y 0 := at a point. Then Y identically coincides with Y 0. Since d(x 1 ω 0 1+X 2 ω 0 2 ) = 0, we can locally find an R 4 -valued function γ satisfying dγ = X 1 ω X 2 ω 0. 2 In addition, γ, ẽ 1 := X 1, ẽ 2 := X 2, ι := X 3, ν := X 4 satisfy (3.3). Suppose L 0 0 and let Ω be a 5 5 matrix such that the components are 1-forms on D as in the right side of (3.3). Then Ω satisfies dω+ω Ω = 0. Therefore we have a unique local solution X =(X 1 X 2 X 3 X 4 X 5 )of 16

17 dx = XΩ for a given initial value, where each X i is an R 5 -valued function. We set L 0 / L T := Then Y := t XT X satisfies dy = t ΩY + Y Ω. Suppose that Y is equal to 1/L Y := at a point. Then Y identically coincides with Y. We see that γ := X 1, ẽ 1 := X 2, ẽ 2 := X 3, ι := X 4, ν := X 5 satisfy (3.3). Hence there exists a neighborhood U of each a D satisfying γ : U N 4 1 is a space-like immersion with zero mean curvature vector, the induced metric by γ is given by g = e 2φ (du 2 + dv 2 ) and if θ (1/4+n/2)π for any n Z, then the curvature K with respect to g is nowhere equal to L 0, ι, ν are light-like normal vector fields of γ : U N1 4 satisfying ι, ν = 1, and principal curvatures of γ with respect to ι, ν at each point of U are given by ±1, ±k, respectively. Therefore we obtain Theorem 1.3. Remark For θ 0 (π/2)z, let φ be a local solution of (1.5) with θ := θ 0. Then noticing (3.3), we see that (cos 2θ 0 )e 2φ ι + e 2φ ν is constant, and this vector is space-like (respectively, time-like) if cos 2θ 0 = 1 (respectively, 1). This means that for the above φ and θ θ 0, the surface is contained in a totally geodesic hypersurface ˆN of N1 4.IfL 0 = 0, then ˆN is isometric to either E 3 or the Minkowski 3-space E1 3;ifL 0 = 1, then ˆN is isometric to either S 3 or the de Sitter 3-space S1 3;ifL 0 = 1, then ˆN is isometric to either H 3 or the anti-de Sitter 3-space H1 3. References [1] L. J. Alías and B. Palmer, Curvature properties of zero mean curvature surfaces in four-dimensional Lorenzian space forms, Mathematical Proceedings of the Cambridge Philosophical Society 124 (1998)

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19 [18] T. Itoh, Minimal surfaces in 4-dimensional Riemannian manifolds of constant curvature, Kodai Math. Sem. Rep. 23 (1971) [19] S. Montiel, Willmore two-spheres in the four-sphere, Trans. Amer. Math. Soc. 352 (2000) [20] B. O Neill, Semi-Riemannian geometry, With applications to relativity, Pure and Applied Mathematics 103, Academic Press, Inc., New York, [21] M. Spivak, A comprehensive introduction to differential geometry, vol. 4, Publish or Perish, [22] R. A. Tribuzy and I. V. Guadalupe, Minimal immersions of surfaces into 4- dimensional space forms, Rendiconti del Seminario Matematico della Universitá di Padova 73 (1985) Graduate School of Science and Technology, Kumamoto University Kurokami, Kumamoto Japan address: ando@sci.kumamoto-u.ac.jp 19