Molding surfaces and Liouville s equation

Transcription

1 Molding surfaces and Liouville s equation Naoya Ando Abstract According to [9], a molding surface has a property that a family of lines of curvature consists of geodesics. In the present paper, we will prove this by another way: our main object of study is the compatibility condition of an over-determined system in relation to Liouville s equation. In addition, we will explicitly describe the solutions of over-determined systems of the same type with the compatibility condition. 1 Introduction A surface S in E 3 with no umbilical points and nowhere zero Gaussian curvature is said to be molding if S belongs to a continuous family of surfaces which are connected by isometries preserving principal distributions but not congruent in E 3 with one another. A molding surface has the following remarkable property: a family of lines of curvature consists of geodesics. This is never trivial. In [10, pp ], this was stated and in [9, pp ], this was provided with a careful proof. The first purpose of the present paper is to carry out another proof, based on the discussion in [3]. Referring to [3], we see that for a principal curvature k of S, F := log k 2 is a solution of the following over-determined system: F u = α + βe F, F v = γ + δe F, (1.1) where α := 2(log K B) u, β := 2 (log B) u K, γ := 2(log A) v, δ := 2K(log A) v, (1.2) and (i) (u, v) are local coordinates on a neighborhood U of each point of S such that / u, / v give principal directions at any point, (ii) A, B are smooth, positive-valued functions on U such that the first fundamental form g of S is locally represented as g = A 2 du 2 + B 2 dv 2, (iii) K is the Gaussian curvature of S. 1

2 The set of solutions of (1.1) is determined by the first fundamental form and a pair of principal distributions. The condition in the above definition of molding surfaces is equivalent to the compatibility condition β v + βγ =0, α v γ u +2βδ =0, δ u + αδ = 0 (1.3) of an over-determined system (1.1). Suppose that S is molding. Notice that the condition βδ 0 is equivalent to the condition that a family of lines of curvature of S consists of geodesics. If we suppose βδ 0 and in addition, if we suppose βδ < 0, then we see from (1.3) that ψ := log 2βδ is a solution of Liouville s equation: ψ uv = e ψ. In Section 3, we will show that ψ uv = e ψ with ψ = log 2βδ causes a contradiction and therefore we will reach βδ 0. Remark Whether S satisfies (1.3) is determined by the first fundamental form and a pair of principal distributions. Our proof in Section 3 needs only to study a two-dimensional Riemannian manifold equipped with an orthogonal pair of two one-dimensional distributions which is connected with the metric by (1.3), while the proof in [9] also needs principal curvatures. In the appendix below, we will describe an outline of the proof in [9]. Both of our proof in Section 3 and the proof in [9] are suitable only for the case where the ambient space is flat: in the case where the ambient space is a 3-dimensional non-flat space form, the compatibility condition of over-determined systems on surfaces does not necessarily imply the existence of a family of geodesics of curvature (see [8]). Remark Molding surfaces are closely related to parallel curved surfaces. Parallel curved surfaces were studied in [1], [2], [3]. A surface S in E 3 is said to be parallel curved if there exists a plane P such that at each point of S, at least one principal direction is parallel to P ;ifs is parallel curved, then such a plane as P is called a base plane of S. For example, a surface of revolution is parallel curved and a plane orthogonal to an axis of rotation is a base plane of a surface of revolution. A canonical parallel curved surface is represented as a disjoint union of plane curves which are congruent in E 3 with one another and tangent to principal directions. These curves are lines of curvature and geodesics. For a surface S in E 3 with no umbilical points and nowhere zero Gaussian curvature, S is molding if and only if a neighborhood of each point of S is a canonical parallel curved surface: the latter holds if and only if S satisfies (a) the compatibility condition (1.3), (b) a family of lines of curvature consists of geodesics (see [3]); condition (b) can be removed, because (a) implies (b). Remark Even if an over-determined system (1.1) does not satisfy the compatibility condition (1.3), it is possible that (1.1) has plural solutions, and in such a case, (1.1) has just 2

3 two solutions. An over-determined system in the form of (1.1) with just two solutions was studied in [6]. We see that F is a solution of (1.1) if and only if F := F + c satisfies where c satisfies c v = γ, and F u = α + β e F, F v = δ e F, (1.4) α := α + c u, β := βe c, δ := δe c. (1.5) An over-determined system in the form of (1.4) has just two solutions if and only if there exist a smooth function p of two variables u, v and a smooth, positive-valued function q of one variable u satisfying p(u, v) >q(u), ( ( )) p α 2 q 2 = log, β = 1 ( ( )) p 2 q 2 log, δ = p q 2p q 2 v u and β v 0 ([6]). In addition, (1.1) with (1.2) has just two solutions if and only if we can choose (u, v) so that A, B satisfy B u /A = w u, A v /B = w v for a smooth nowhere zero function w satisfying (log tanh w ) uv 0 and the sinh-gordon equation w uu + w vv = sinh 2w ([6]). See [8] for an over-determined system on a surface in a 3-dimensional non-flat space form with just two solutions. In the case where we can choose isothermal coordinates as (u, v), the surface has nonzero constant mean curvature. u Remark We set X := β v, Y := α v +2β δ, Z := δ u + α δ. An over-determined system in the form of (1.4) has a unique solution so that Y 2 =4XZ holds if and only if there exist a smooth function Δ of two variables u, v and a smooth function f of one variable u satisfying α = f(u) + 2(log Δ) u, β = f(u) ( ) 1 Δ +, δ =Δ v Δ u and (log Δ) uv ((log Δ) u f(u))(log Δ) v ([5]). In addition, (1.1) with (1.2) has a unique solution so that Y 2 =4XZ holds if and only if we can choose (u, v) so that A, B satisfy B u /A = w u, A v /B = w v for a smooth function w satisfying w uv +2w u w v 0 and w uu + w vv = e 2w ([5]). See [8] for an over-determined system on a surface in a 3-dimensional non-flat space form with a unique solution and Y 2 =4XZ. In the case where we can choose isothermal coordinates as (u, v), the surface is minimal. Remark We know the general solution ψ of Liouville s equation ψ uv = e ψ : e ψ = 2S (u)t (v) (S(u)+T (v)) 2, 3

4 where S, T are smooth functions of one variable. See [12, pp ], [11, p. 159] for the method by a Bäcklund transformation, and see [6] for the method by studying an over-determined system in the form of (1.1) with the compatibility condition. Let α, β, γ, δ be smooth functions on a simply connected domain of R 2 which are not necessarily represented as in (1.2) by A, B and K. If they satisfy (1.3), then for an arbitrarily given initial value at a point, there exists a unique solution of (1.1) with the initial value. The second purpose of the present paper is to describe the solution explicitly. (a) If β 0 and if δ 0, then (1.3) implies that α, γ satisfy α v = γ u, and therefore a unique solution F of (1.1) with an initial value F 0 at a point (u 0,v 0 ) is given by F (u, v) =F 0 + (u,v) (u 0,v 0 ) αdu + γdv. (b) Suppose β 0 and δ>0. Then from (1.3), we see that there exists a smooth function t of one variable v satisfying γ (log δ) v = t(v). Let T be a primitive function of t, so that T = t holds. Then a unique solution F of (1.1) with an initial value F 0 at (u 0,v 0 ) is given by ( v ) F (u, v) = log δ(u, v)+t (v) + log e F 0 + e T (v) dv, v 0 where we suppose δ(u 0,v 0 ) = 1 and T (v 0 )=0. (c) Suppose β<0and δ>0. Since log 2βδ is a solution of Liouville s equation, there exist a smooth function S of u and a smooth function T of v satisfying βδ = S (u)t (v) (S(u)+T(v)). 2 Then a unique solution F of (1.1) with an initial value F 0 at (u 0,v 0 ) is given by e F (u,v) S (u)(c T (v)) = (S(u)+C)(S(u)+T(v))β(u, v) (S(u)+T(v))(T (v) C)δ(u, v) =, (S(u)+C)T (v) where C := S 0T 0 + S 0 (S 0 + T 0 )β 0 e F 0 S 0 (S = (S 0 + T 0 )T 0 δ 0 S 0 T 0e F T 0 )β 0 e F 0 (S 0 + T 0 )δ 0 + T 0 ef 0 and S 0 := S(u 0 ), T 0 := T (v 0 ) and so on. Remark For a molding surface, functions β, δ as in (1.2) satisfy βδ 0. In addition, either β 0orδ 0 holds: if both β 0 and δ 0 hold, then we obtain K 0 and this causes a contradiction. Therefore we can consider a pair (β,δ) to be of type (b) in the last paragraph. 4

5 2 Preliminaries Let M be a two-dimensional Riemannian manifold and g its metric. Let D 1, D 2 be two smooth one-dimensional distributions on M. Then (M,g) equipped with (D 1, D 2 ) is called a semisurface if D 1 and D 2 are orthogonal to each other at any point of M with respect to g. If (M,g, D 1, D 2 ) is a semisurface, then a triplet (g, D 1, D 2 ) is called a semisurface structure of M. The canonical pre-divergence V K of (M,g, D 1, D 2 ) is a vector field on M defined by the sum of the geodesic curvature vectors of the two integral curves of D 1 and D 2 through each point of M. Let ι : M E 3 be an isometric immersion of M into E 3. Suppose that M has no umbilical points with respect to ι. Let D 1, D 2 be principal distributions on M with respect to ι which give the two principal directions at each point of M. Then g, D 1 and D 2 form a semisurface structure of M. Let k 1, k 2 be principal curvatures of M with respect to ι corresponding to D 1, D 2, respectively. Then we have the equation of Gauss div (V K )=k 1 k 2, (2.1) where div (V K ) is the divergence of V K, and the equations of Codazzi-Mainardi U 1 (k 2 )U 1 + U 2 (k 1 )U 2 =(k 1 k 2 )V K, (2.2) where U 1, U 2 are smooth unit vector fields on a neighborhood of each point of M satisfying U i D i ([7]). Let (u, v), A, B be as in Section 1. Then we can rewrite (2.2) into (k 1 ) v = (log A) v (k 1 k 2 ), (k 2 ) u = (log B) u (k 1 k 2 ). (2.3) Suppose that the curvature K of (M,g) is nowhere zero. Then by div (V K )=K, (2.1) and (2.3), we see that F := log k1 2 is a solution of (1.1) with (1.2). In addition, from F uv = F vu, we obtain P (k 1,k 2 ) = 0, where P is a homogeneous polynomial of degree two in two variables defined by P (X 1,X 2 ) := 1 { K(β v + βγ)x1 2 +(α v γ u +2βδ)X 1 X 2 + δ } u + αδ X2 2. 2AB K The compatibility condition (1.3) is equivalent to P q 0 for any point q of M. We see that P is determined by the semisurface structure (g, D 1, D 2 ) at each point of M and does not depend on the choice of (u, v) up to a sign. We call P the Codazzi-Mainardi polynomial of (M,g, D 1, D 2 ). We can represent P by the geodesic curvatures of integral curves of D 1 and D 2 ([7]). If P 0, then we can represent each of k 1 and k 2 by the coefficients of P and K (for concrete representations, see [3]). If P q 0 for any point q of M, then the image of M by ι is a molding surface: more strictly, we see that for each point q M and each pair of two nonzero numbers (k (0) 1,k (0) 2 ) satisfying k (0) 1 k (0) 2 = K(q), there exists an isometric immersion ι of a neighborhood of q into E 3 satisfying 5

6 (a) (D 1, D 2 ) gives a pair of two principal distributions, (b) the principal curvatures at q corresponding to D 1, D 2 are given by k (0) 1, k(0) 2, respectively; such an immersion as ι is uniquely determined by (k (0) 1,k(0) 2 ) up to an isometry of E3 ([3]). Conversely, for a molding surface, the Codazzi-Mainardi polynomial vanishes at any point. Therefore we can attain the first purpose of the present paper, by proving Theorem 2.1 Let (M,g, D 1, D 2 ) be a semisurface with nowhere zero curvature satisfying P q 0 for any point q of M. Then the integral curves of one of D 1 and D 2 are geodesics. Remark Let Φ, Ψ be smooth functions of u, v, w on a domain D of R 3. Let F be a solution of an over-determined system F u =Φ(u, v, F ), F v =Ψ(u, v, F ). (2.4) Then by F uv = F vu, we obtain Φ v +Φ w Ψ=Ψ u +Ψ w Φat(u, v, F (u, v)) for any (u, v). The compatibility condition of (2.4) is given by Φ v +Φ w Ψ Ψ u +Ψ w ΦonD. If (2.4) satisfies the compatibility condition, then for each (u 0,v 0,w 0 ) D, there exists a unique solution F of (2.4) on a neighborhood of (u 0,v 0 )inr 2 satisfying F (u 0,v 0 )=w 0 ([13, p. 393]). Let θ be a 1-form on D defined by θ := Φdu +Ψdv dw. Then a Pfaffian equation θ =0is completely integrable if and only if (2.4) satisfies the compatibility condition. Let α, β, γ, δ be smooth functions of u, v. If we set Φ(u, v, w) :=α(u, v)+β(u, v)e w, Ψ(u, v, w) :=γ(u, v)+δ(u, v)e w, then (2.4) is represented as (1.1) and the compatibility condition Φ v +Φ w Ψ Ψ u +Ψ w Φ is represented as (1.3). Remark There exists a surface in E 3 with a family of geodesics of curvature which is not parallel curved. In [4], the lines of curvature of such a surface are characterized intrinsically and extrinsically: the semisurface structure of such a surface is characterized in terms of local representation of the first fundamental form; the curvatures and the torsions of the lines of curvature as space curves are characterized. In [7], we can find rewrites of theorems in [4] with respect to the intrinsic characterization of such a surface as above, in terms of the geodesic curvatures of lines of curvature. 3 Proof of Theorem 2.1 Suppose βδ 0. Then we can suppose β<0and δ>0. Then the first relation in (1.3) implies that there exists a smooth, positive-valued function x of one variable u satisfying 6

7 KA 2 = 2(log B) u x(u); the third relation in (1.3) implies that there exists a smooth, positive-valued function y of one variable v satisfying KB 2 = 2(log A) v y(v). Therefore we obtain K =2 (log B) ux(u) A 2 =2 (log A) vy(v) B 2. (3.1) Let ũ be a smooth function of one variable u satisfying (dũ/du)(u) =x(u) and ṽ a smooth function of one variable v satisfying (dṽ/dv)(v) = y(v). Then (ũ, ṽ) are local coordinates satisfying / ũ D 1, / ṽ D 2 and we can rewrite (3.1) into (log B)ũ K =2 à 2 =2 (log Ã) ṽ B 2, (3.2) where à := A/x and B := B/y. Then we obtain Bũ/à = Ãṽ/ B. In the following, we will use u, v, A, B, instead of ũ, ṽ, Ã, B, respectively. The following hold: K = 1 {( ) ( ) } Av Bu + = X u + X v AB B A AB, (3.3) v where X := A v /B = B u /A. By (3.2) together with (3.3), we obtain X u + X v = 2X. This implies X = exp( u v + φ(u v)), where φ is a smooth function of one variable. Therefore we obtain 2βδ = 8X 2 = 8 exp 2( u v + φ(u v)). If we set ψ := log 2βδ, then we obtain ψ uv = 2φ (u v) and therefore we obtain ψ uv e ψ. On the other hand, we see from (1.3) that ψ must satisfy ψ uv = e ψ. Hence we have a contradiction and therefore we obtain βδ 0. Hence we have proved Theorem 2.1. Remark We set I := B u /A, J := A v /B. Then α, β, δ in (1.5) with (1.2) and c = 2 log A are represented as α = (log(i u + J v ) 2 ) u, β = 2I, δ = 2(I u + J v )J. (3.4) I u + J v Suppose that M is oriented. Let θ K be a 1-form on M corresponding to V K by g and set ω := θ K, where is Hodge s -operator. Then ω is represented as ω = Jdu + Idv. The condition K 0 is equivalent to dω 0. Whether (1.4) with (3.4) has a solution is determined by (ω, D 1, D 2 ), and a condition of the existence of a solution of (1.4) with (3.4) is represented as a relation among ω, D 1, D 2 ([5]). Whether (1.4) with (3.4) satisfies the compatibility condition is determined by (ω, D 1, D 2 ), and referring to the above proof of Theorem 2.1, we can show that if (1.4) with (3.4) satisfies the compatibility condition, then IJ 0. 7 u

8 A Appendix: An outline of the proof in [9] Let M, ι be as in the beginning of Section 2. Let e 1, e 2 form an orthonormal frame field on a neighborhood U p of each point p of M. Let e 3 be a unit normal vector field on U p with respect to ι. We set 3 dι = ω 1 e 1 + ω 2 e 2, de i = ω ij e j, where ω i and ω ij are 1-forms on U p. Then we have ω ij + ω ji = 0 and the structural equations: 3 dω 1 = ω 12 ω 2, dω 2 = ω 1 ω 12, dω ij = ω ik ω kj. We set ω 12 := l 1 ω 1 + l 2 ω 2. Then l i is the geodesic curvature of each integral curve of e i for i {1, 2}. Suppose that each e i gives a principal distribution on U p and let k i be the principal curvature corresponding to e i. Then we have ω i3 = k i ω i. Suppose K 0 and set Q i := l i, π 1 := Q 2 ω 13, π 2 := Q 1 ω 23. k i In the following, suppose that ι(m) is a molding surface. Then we can choose U p so that for each t 0 R \{1, 0, 1}, there exists a smooth function t on U p with t(p) =t 0 satisfying t 1, 0, 1 and ( e 2 (tk 1 )= tk 1 k ) ( ) ( 2 k2 l 1, e 1 = tk 1 k ) 2 l 2 t t t on U p. These imply 1 2 d log 1 t2 = t 2 π 1 π 2. Computing the exterior differentiations of both sides, we obtain j=1 t 2 (dπ 1 2π 1 π 2 )=dπ 2 2π 1 π 2. Since t 0 R \{1, 0, 1} can be chosen arbitrarily, we obtain dπ 1 = dπ 2 =2π 1 π 2. These imply that there exist smooth functions R 1, R 2 satisfying dq 1 = Q 1 Q 2 ω 13 + R 2 ω 23, dq 2 = R 1 ω 13 Q 1 Q 2 ω 23. (A.1) Noticing ω 12 = Q 1 ω 13 + Q 2 ω 23 and computing the exterior differentiations of both sides of this, we obtain R 1 R 2 +1+Q Q 2 2 =0. By this together with (A.1), we obtain k=1 dr 1 = 2(2Q R 1 )Q 2 ω 13 2Q 1 Q 2 2ω 23, dr 2 = 2Q 2 1 Q 2ω 13 2Q 1 (2Q 2 2 R 2)ω 23. By these together with ddr i = 0, we obtain Q 1 Q 2 = 0. This implies l 1 l 2 =0. 8

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