aa r X i v : . [ m a t h . DG ] A ug DEGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE
JOE S. WANG
Abstract.
We give a local analytic characterization that a minimal surface in the 3-sphere S ⊂ R defined by an irreducible cubic polynomial is one of the Lawson’s minimal tori. This provides an alternativeproof of the result by Perdomo ( Characterization of order 3 algebraic immersed minimal surfaces of S ,Geom. Dedicata 129 (2007), 23–34). Contents
1. Introduction 12. Minimal surfaces in S h − h + h h − = 0 144.3. Case h − h + h h − = 0 15Appendix. 17References 191. Introduction
Let E = R be the four dimensional Euclidean vector space. Let S ⊂ E be the unit sphere equippedwith the induced Riemannian metric. A minimal surface M ֒ → S is by definition an immersed surfacewith vanishing mean curvature, which locally gives an extremal for the area functional. The differentialequation for minimal surfaces is an elliptic Monge-Ampere equation defined on the unit tangent bundleof S . It is well known that the minimal surface equation is locally equivalent to the elliptic sinh-Gordonequation for one scalar function of two variables;(1.1) ∆ u + sinh u = 0 . Given a surface
M ֒ → S , consider the cone O ∗ M ֒ → E over the origin O ∈ E . Then M ֒ → S isminimal whenever O ∗ M ֒ → E is minimal as a hypersurface in the Euclidean space E . In this respect,a minimal surface M ֒ → S is called algebraic of degree m if there exists a nonzero irreducible degree m homogeneous polynomial F : E → R which vanishes on M . A nonzero irreducible homogeneouspolynomial F defines a (possibly singular) minimal surface when(1.2) |∇ F | ∆ F − ( ∇ F ) t · H ( F ) · ( ∇ F ) ≡ , mod F. Mathematics Subject Classification.
Key words and phrases. algebraic minimal surface, 3-sphere, cubic polynomial.
Here ∇ F, ∆ F, H ( F ) denote the gradient, the Laplacian, and the Hessian of the function F respectivelywith respect to the Euclidean metric of E , [Hs, p260].Hsiang classified the homogeneous minimal hypersurfaces in the standard Euclidean spheres, [Hs]. In[Hs], he also showed that the totally geodesic 2-sphere and Clifford torus are the only algebraic minimalsurfaces in S of degree ≤
2. Lawson constructed an infinite sequence of algebraic minimal tori(orKlein bottles) in S of arbitrary high degree, [La]. Hsiang and Lawson gave an analysis for the minimalsubmanifolds of low cohomogeneity in general Riemannian homogeneous spaces, where, in particular,the Lawson’s sequence of algebraic minimal tori are extended to a countable family of cohomogeneity 1minimal tori in S , [HsL, p32].Recently, Perdomo gave a characterization of degree 3 algebraic minimal surfaces in S as one of theLawson’s algebraic minimal tori, [Pe1, Pe2]. One of the main idea of his analysis is that such a minimalsurface necessarily contains a great circle, which puts the defining cubic polynomial into a special normalform. The minimal surface equation (1.2) is then applied to successively normalize the polynomialcoefficients by differential algebraic analysis.The purpose of the present paper is to give an analytic characterization of degree 3 algebraic minimalsurfaces in S . We employ the method of local differential analysis and show that Lawson’s algebraicminimal torus of degree 3 is the only minimal surface in S which satisfies the compatibility equation tolie in the zero locus of an irreducible cubic polynomial. Main results.
1. The structure equation for the degree 3 algebraic minimal surfaces in S is determined, Theorem4.1. The analysis for the structure equation provides the formula for the defining cubic polynomial, andit is explicitly identified as the Lawson’s degree 3 example.2. The structure equation shows that the degree 3 algebraic minimal surface is the conjugate surfaceof a principally bi-planar minimal surface. It has the Killing nullity 1, and the curvature takes valuesin the closed interval [ − , ].Let us give an outline of the analysis. The problem of finding an algebraic minimal surface in S canbe interpreted as the problem of finding a solution to the sinh-Gordon equation (1.1) with the propertythat it admits an associated, in a certain algebraic manner, polynomial F which satisfies (1.2). This isequivalent to finding a constant section of an appropriate tensor bundle over the minimal surface whichvanishes when evaluated on the surface. As the degree of F increases, this problem imposes a sequenceof compatibility equations on the higher order structure functions of the minimal surface. For the degree3 case treated in this paper, the compatibility equations are reduced to a pair of third order equations. Main results are obtained by applying the over-determined PDE analysis to these equations.The present work can be considered as a coordinate free and equivariant interpretation of the afore-mentioned Perdomo’s original characterization. Compared to his analysis, one possible advantage wouldbe that this interpretation of an algebraic minimal surface is susceptible to local analytic method, andcould be, in theory, applied to general higher degree cases. However, it is practically difficult to performthe required computations manually. Our analysis relies essentially on the use of computer machine.On the other hand, our analysis also suggests as a byproduct a distinguished class of minimal surfacesdefined by an additional fourth order equation ( ∆ = 0 in (4.8)). An analysis of this class of minimalsurfaces will be reported in the subsequent paper.We carried out the analysis for the degree 4 algebraic minimal surfaces by the same method used inthis paper. The analysis indicates that such a minimal surface is one of the family of cohomogeneity 1minimal tori described by Hsiang and Lawson. Partly due to the length and complexity of the algebraicanalysis involved, we do not present the details of the analysis for degree 4 case in this paper. A minimal surface in S is principally bi-planar when each of its principal curves lies in a totally geodesic S ⊂ S ,[Ya]. There exist locally one parameter family of distinct principally bi-planar minimal surfaces in S . The analysis of the degree 4 algebraic minimal surfaces was the original motivation for the present work. One needs acharacterization of the degree 3 surfaces first to exclude them from the analysis for the degree 4 surfaces.
EGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE 3
In a different context, similar analysis can be applied to special Legendrian surfaces in the 5-sphere,which are the links of special Lagrangian 3-folds in C , [Ha]. A special Legendrian surface is called quasi-algebraic of degree m if it lies in the zero locus of a nonzero irreducible degree m real homogeneouspolynomial F : C → R . The analysis shows that;1. a quasi-algebraic degree 2 special Legendrian surface is necessarily one of the cohomogeneity 1surfaces treated by Haskins, [Ha],2. a quasi-algebraic degree 3 special Legendrian surface with an appropriate Z symmetry is necessarilya quasi-algebraic surface of degree 2.The computation indicates in fact that any quasi-algebraic degree 3 special Legendrian surface isnecessarily a quasi-algebraic surface of degree 2. We do not have a complete proof of this claim at thistime of writing.The paper is organized as follows. In Section 2, we set the basic structure equation for the minimalsurfaces in S , and compute its infinite sequence of prolongations, (2.8), Lemma 2.14. In order to simplifythe computations, the induced complex structure on the minimal surface is utilized, and the structureequations are written in complex form. In Section 3, we give a description of the method to detect thecompatibility equations for a minimal surface to be algebraic. The method is applied to two simplecases of degree 1, and degree 2 algebraic minimal surfaces, Theorem 3.6, Theorem 3.8. In Section 4, wepresent the results of differential analysis for degree 3 algebraic minimal surfaces, Theorem 4.1. Aftera preliminary reduction in Section 4.1, the analysis is divided into two cases. The differential analysisshows that a degree 3 algebraic minimal surface necessarily satisfies an auxiliary equation either of order3, Section 4.2, or of order 4, Section 4.3. The auxiliary equation of order 3 is compatible and supports adefining irreducible cubic polynomial, which is explicitly verified as the one given by Lawson.The majority of computations were performed using the computer algebra system Maple with difforms package. Throughout the paper, a surface is a connected smooth two dimensional manifold.2.
Minimal surfaces in S In this section, we set the basic structure equations for a minimal surface in S . In Section 2.1, weapply the moving frame method to determine the structure equation for an immersed oriented minimalsurface in S . In Section 2.2, we compute the infinite prolongation of the structure equation explicitly,Lemma 2.14.The analysis and results in this section are classical, and well known. For the standard reference onthe theory of minimal surfaces in S , we refer to [La] and the references therein.The basic structure equations established in this section will be used implicitly throughout the paper.2.1. Structure equation.
Let E = R be the four dimensional Euclidean vector space. Let S ⊂ E bethe unit sphere equipped with the induced Riemannian metric. The special orthogonal group SO actstransitively on S as a group of isometry, and S = SO / SO .Let Λ → S be the S -bundle of unit tangent vectors. Let Gr + (2 , E ) be the Grassmannian of twodimensional oriented subspaces of E . SO acts transitively on both Λ and Gr + (2 , E ), and there existsthe incidence double fibration; The
Maple worksheet is available upon request by email.
JOE S. WANG SO π ↓ Λ = SO / SO ցւ Gr + (2 , E ) = SO / (SO × SO )S π π Figure 2.1. Double fibrationTo fix the notation once and for all, let us define the projection maps π, π , and π explicitly. Let e = ( e , e , e , e ) denote the SO ⊂ GL R frame of E . Define π ( e ) = ( e , e ∧ e ) , (2.1) π ( e , e ∧ e ) = e ,π ( e , e ∧ e ) = e ∧ e . The SO -frame e satisfies the structure equation de A = X B e B ω BA , (2.2) ω AB + ω BA = 0 , for the Maurer-Cartan form ( ω AB ) of SO . ( ω AB ) satisfies the compatibility equation(2.3) dω AB + X C ω AC ∧ ω CB = 0 . Let x :
M ֒ → S be an immersed oriented surface. By the general theory of moving frames, there existsa lift ˜x : M ֒ → Λ such that e = x, and e is the oriented normal to the surface x. Let ˜x ∗ SO → M bethe pulled back SO -bundle. We continue to use ( ω AB ) to denote the pulled back Maurer-Cartan formon ˜x ∗ SO . From (2.1), (2.2), the initial state of ( ω AB ) on ˜x ∗ SO takes the form(2.4) ( ω AB ) = · − ω − ω · ω · ω ω ω ω · ω · ω ω · . Here ’ · ’ denotes zero, and ω A = ω A , A = 1 ,
2. By definition, I = h de , de i = ( ω ) + ( ω ) is theinduced Riemannian metric of the immersed surface, where h , i is the inner product of E .Differentiating ω = 0, one gets ω ∧ ω + ω ∧ ω = 0 . By Cartan’s lemma, there exist coefficients h AB , A, B = 1 ,
2, symmetric in indices such that ω A = X B h AB ω B . The structure equation shows that the quadratic differential(2.5) II = ω A ◦ ω A = h AB ω A ◦ ω B is well defined on M . II is the second fundamental form of the immersed surface x. Definition 2.6.
Let x :
M ֒ → S be an immersed oriented surface. Let II be the quadratic differential (2.5) which is the second fundamental form of the immersed surface x . x is minimal when the trace ofthe quadratic differential II with respect to the induced metric I vanishes, or equivalently when h + h = 0 . EGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE 5
From now on, a minimal surface would mean an immersed oriented minimal surface.In order to utilize the induced complex structure on M as a Riemann surface, let us introduce thecomplexified structure equation. Let E C = E ⊗ C be the complexification, and consider the following E C -frame. e C = ( e , E , E − , e ) , (2.7) E = 12 ( e − i e ) , i = − ,E − = E . Rewriting (2.2), (2.4) with respect to e C , one gets(2.8) de C = e C · − ω − ω · ω − i ρ · − h ωω · i ρ − h ω · h ω h ω · , where we set ω = ω + i ω ,ρ = ω ,h = h − i h . Differentiating (2.8), one gets the compatibility equations dω = i ρ ∧ ω, (2.9) dρ = K i2 ω ∧ ω, where K = 1 − h h ,dh + 2 i h ρ ≡ , mod ω. Here K is the curvature of the induced metric on the minimal surface. Remark 2.10.
The subscript ’2’ in the notation ’ h ’ represents the weight of the action by the structuregroup SO . It is convenient for a computational purpose. (2.9) shows that the quadratic differential(2.11) II C = h ω ◦ ω is well defined on M , and is holomorphic with respect to the complex structure on M defined by the(1 , ω . II C is the complexified second fundamental form of the minimal surface. Definition 2.12.
Let x :
M ֒ → S be an immersed oriented minimal surface. The holomorphic quadraticdifferential II C , (2.11) , is the Hopf differential of the minimal surface. The zero set of II C is the umbilicdivisor . Example 2.13.
Consider an immersed minimal sphere S ֒ → S . Since S supports no nonzero holo-morphic differentials of positive degree, II C = 0 and the structure coefficient h vanishes identically. Aminimal sphere in S is necessarily totally geodesic.Conversely, it is clear from the structure equation (2.8) that a minimal surface in S with h ≡ islocally equivalent to the totally geodesic sphere. For a compact minimal surface
M ֒ → S of genus g ≥
1, the umbilic divisor has degree 4 g − JOE S. WANG
Prolongation.
In this section, we compute the infinite sequence of prolongations of the structureequation (2.9). The prolonged structure equation will be used implicitly for the differential analysis inSection 4.Let us introduce the higher order derivatives of the structure coefficient h in (2.9) inductively by dh j + i jh j ρ = h j +1 ω + h j, − ω, j = 2 , , ... . For a notational purpose, denote h − j = h j , j = 2 , , ... . For instance, the curvature of the surface iswritten as K = 1 − h h = 1 − h h − . Lemma 2.14. h , − = 0 ,h j +1 , − = j − X s =0 c js h j − s ∂ s K, for j ≥ , where c js = ( j + s + 2)2 ( j − j − s − s + 2)! = ( j + s + 2)2 j (cid:18) js + 2 (cid:19) . Here by definition ∂ s K = δ s − h s h − for s ≥ .Proof. Differentiating dh j and collecting ω ∧ ω -terms, one gets h j +1 , − = ∂h j, − + j h j K, where ∂h j, − is the ω -component of dh j, − ( ∂ s K is defined similarly). The formula is verified by directcomputation. Note that a j = ( j + 2)( j − ,a j ( j − = 1 . (cid:3) Example 2.15.
Consider R = C ⊕ C = C . Clifford torus is the minimal surface in S defined as theproduct of circles { ( z , z ) ∈ C | | z | = | z | = 12 } . The induced Riemannian metric is flat, and the curvature K vanishes. Lemma 2.14 shows that ∂K = − h h − . Clifford torus is not totally geodesic, and this implies h ≡ .Conversely, consider a minimal surface in S such that the structure function h ≡ . By Lemma2.14, ∂h = h K ≡ , and either h ≡ (and K = 1 ), or K = 0 . In the latter case, it is well knownthat the surface is locally congruent to Clifford torus. Algebraic minimal surfaces
Let S m ( E ) be the vector space of real, homogeneous degree m polynomials on the four dimensionalEuclidean vector space E . By the metric duality, we identify S m ( E ) with the space of symmetric m -tensors Sym m ( E ). Let S ( E ) = ⊕ ∞ m =0 S m ( E ). Definition 3.1.
Let x :
M ֒ → S ⊂ E be a minimal surface in the unit sphere. x is algebraic ifthere exists a nonzero homogeneous polynomial F ∈ S ( E ) which vanishes on x . For each m ≥ , let J m x ⊂ S m ( E ) be the subspace of degree m polynomials vanishing on x . Degree of an algebraic minimalsurface x is the minimum integer m such that J m x is nontrivial. EGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE 7
Note by definition that for an algebraic minimal surface x of degree m , J m x is nontrivial for all m ≥ m .In this section, we first describe the idea of how the over-determined PDE analysis can be applied todetect the compatibility equations for a minimal surface to be algebraic, Section 3.1. We then apply thisto the cases of degree 1, and degree 2 algebraic minimal surfaces, Section 3.2, Section 3.3.We continue to use the structure equations established in Section 2. The method of local differentialanalysis described in this section will be applied to degree 3 algebraic minimal surfaces in Section 4.3.1. Structure equation.
Let x :
M ֒ → S be a minimal surface with the associated complexified frame e C , (2.7). Consider the trivial bundle S m ( E ) × S → S , and the induced bundle x ∗ ( S m ( E ) × S ) → M .A section of x ∗ ( S m ( E ) × S ) over M can be represented in terms of the frame e C as follows. F = i + j + k + l = m X i, j, k, l ≥ p ijkl e i e j E k E l − , p ijkl = p ijlk , for a set of coefficients p ijkl . Equivalently, such F is an S m ( E )-valued function on M .The condition that F is a constant section, or F ∈ S m ( E ) with an abuse of notation, is expressed by(3.2) dF = 0 . Expanding the exterior derivative dF by Leibniz rule in the variables { e , e , E , E − } , one gets thestructure equation for the coefficients { p ijkl } ;(3.3) dp ijkl = X φ i ′ j ′ k ′ l ′ ijkl p i ′ j ′ k ′ l ′ , where φ i ′ j ′ k ′ l ′ ijkl is determined as a linear combination of the components of the Maurer-Cartan form (2.8). Remark 3.4.
The formulae for { φ i ′ j ′ k ′ l ′ ijkl } will be given explicitly in Section 3.2, Section 3.3, Section 4for the cases m = 1 , , respectively. The condition that the polynomial F vanishes on x is expressed by(3.5) p m = 0 . The idea is then to apply the structure equation (3.3) repeatedly starting from the initial state (3.5) byusing (2.9) and (2.14). As one differentiates, the successive derivatives of (3.5) imply more and morelinear compatibility equations on the polynomial coefficients { p ijkl } , thereby reducing the number ofindependent coefficients.In the case F is irreducible, the reduction process eventually leads to a single independent polynomialcoefficient, and the integrability equations for a minimal surface to support such an irreducible polynomialare expressed as a set of algebraic equations on the structure functions { h ± , h ± , ... } . For the minimalsurface satisfying these integrability equations, the formula for F is obtained by evaluating the polynomialcoefficients at an appropriate generic point of the minimal surface.In the next two sections, we examine the well known cases of degree 1, and degree 2 minimal surfacesfollowing this method just described. These two cases not only serve as exercise for the differential analysisof the more complicated case of degree 3 surfaces in Section 4, but they are also necessary prerequisites.From the remark below Definition 3.1, one needs the characterization of degree 1, and degree 2 algebraicminimal surfaces to exclude them from the analysis for the degree 3 surfaces.3.2. Degree 1 algebraic minimal surfaces.
In this section, we give a characterization of degree 1algebraic minimal surfaces.
Theorem 3.6.
Let x :
M ֒ → S ⊂ E be an algebraic minimal surface of degree 1. Then the structurefunctions satisfy h ± = 0 . The minimal surface is congruent to a part of the totally geodesic sphere.
JOE S. WANG
Proof of theorem is presented below in 3 steps.Let us write a real, homogeneous degree 1 polynomial F ∈ S ( E ) in terms of the frame e C . F = p e + p E + p − E − + p e ∈ S ( E ) . The equation dF = 0 is equivalent to the following set of structure equations for the polynomial coeffi-cients. dp − p − ω − p ω = 0 , (3.7) dp + p h ω + p − h − ω = 0 ,dp − i p ρ + 12 p ω − p h − ω = 0 ,dp − + i p − ρ + 12 p ω − p h ω = 0 . Differentiating the coefficients of F would mean applying this structure equation from now on. Step 1 . Assume the initial condition, p = 0 . By the metric duality, this is equivalent to that F vanishes on the minimal surface x. Step 2 . Differentiating p = 0, one gets p ± = 0 . At this stage, the polynomial is reduced to F = p e . Step 3 . Differentiating p ± = 0, one gets p h ∓ = 0 . • If the structure functions h ± do not vanish identically, p = 0 and the minimal surface does notsupport a nonzero degree 1 polynomial vanishing on the surface. • If h ± = 0 identically, the minimal surface is congruent to a part of the totally geodesic sphere,Example 2.13. The structure equation (3.7) is reduced to dp = 0, which is compatible.Fix a point x on the minimal surface. Choose an orthonormal coordinate { x , x , x , x } of E sothat one has the following identification at x by the metric duality. e = x ,E ± = 12 ( x ∓ i x ) ,e = x . Up to scale, one may assume p = 1. The degree 1 polynomial F is given by F = x . (cid:3) Degree 2 algebraic minimal surfaces.
In this section, we give a characterization of degree 2algebraic minimal surfaces.
Theorem 3.8.
Let x :
M ֒ → S ⊂ E be an algebraic minimal surface of degree 2. Then the structurefunctions satisfy − h h − = 0 ,h ± = 0 . The minimal surface is congruent to a part of Clifford torus. For an orthonormal coordinate { x , x , x , x } of E , the surface is defined by the quadratic polynomial F = x x − x x . EGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE 9
Proof of theorem is presented below in 5 steps.Let us write a real, homogeneous degree 2 polynomial F ∈ S ( E ) in terms of the frame e C . F = p , e + 2 p , e e + p , e + 2 r , e E + 2 r , − e E − + 2 r , e E + 2 r , − e E − + 4 r , E + 8 r , − E E − + 4 r − , − E − ∈ S ( E ) . We implicitly assume the appropriate conjugation relations among the coefficients so that F is real, i.e., r , − = r , , etc. The equation dF = 0 is equivalent to the following set of structure equations for thepolynomial coefficients. dp , − r , − ω − r , ω = 0 , (3.9) dp , + 12 ( − r , − + h r , ) ω + 12 ( − r , + h − r , − ) ω = 0 ,dp , + h r , ω + h − r , − ω = 0 ,dr , − i r , ρ + ( p , − r , − ) ω + ( − h − p , − r , ) ω = 0 ,dr , − + i r , − ρ + ( p , − r , − ) ω + ( − h p , − r − , − ) ω = 0 ,dr , − i r , ρ + ( p , + 2 h r , ) ω + , ( − h − p , + 2 h − r , − ) ω = 0 ,dr , − + i r , − ρ + ( p , + 2 h − r − , − ) ω + ( − h p , + 2 h r , − ) ω = 0 ,dr , − r , ρ + 12 r , ω − h − r , ω = 0 ,dr , − + 14 ( r , − − h r , ) ω + 14 ( r , − h − r , − ) ω = 0 ,dr − , − + 2 i r − , − ρ + 12 r , − ω − h r , − ω = 0 . Differentiating the coefficients of F would mean applying this structure equation from now on. Step 1 . Assume the initial condition, Eq , : p , = 0 . By the metric duality, this is equivalent to that the quadratic polynomial F vanishes on the minimalsurface x. Step 2 . Differentiating p , = 0, one gets Eq , : r , = 0 ,Eq , : r , − = 0 . Step 3 . Differentiating r , ± = 0, one gets Eq , : h − p , + 2 r , = 0 , (3.10) Eq , : r , − = 0 ,Eq , : h p , + 2 r − , − = 0 . One may solve for { r , , r , − , r − , − } from these equations. Step 4 . Differentiating (3.10), one gets Eq , : h r , = 0 , (3.11) Eq , : h − r , − = 0 . Since the minimal surface has degree 2, h ± do not vanish identically, and this implies r , ± = 0. Step 5 . Differentiating (3.11), one finally gets Eq , : h p , = 0 ,Eq , : h − p , = 0 ,Eq , : (1 − h h − ) p , = 0 . As in
Step 4 , h ± does not vanish identically, and p , = 0. At this stage, the quadratic polynomial F is reduced to F = 2 p , ( − e e + h − E + h E − ). • If the curvature (1 − h h − ) does not vanish identically, p , = 0 and the minimal surface does notsupport a nonzero degree 2 polynomial vanishing on the surface. • If 1 − h h − = 0 identically, the minimal surface is congruent to Clifford torus, Example 2.15. Thestructure equation (3.9) is reduced to dp , = 0, which is compatible. By the existence and uniquenesstheorem of ODE, there exists up to scale a unique nonzero quadratic polynomial that vanishes on Cliffordtorus.Fix a point x on Clifford torus. From the compatibility equation 1 − h h − = 0, one may adapt theframe at x so that h ± = 1. Choose an orthonormal coordinate { x , x , x , x } of E so that one hasthe following identification at x by the metric duality. e = x ,E = 12 ( x − i x ) exp (i π ,E − = 12 ( x + i x ) exp ( − i π ,e = x . Up to scale, one may assume p , = − . The degree 2 polynomial F is given by F = x x − x x . (cid:3) Comparing the analysis for degree 1, and degree 2 minimal surfaces, it is evident that the analysisfor higher degree algebraic minimal surfaces would follow the same path, but that the computationalcomplexity would increase due to possibly the large number of terms involving the higher order structurefunctions h ± , h ± , ... . We shall show in the next section that the required differential analysis ismanageable for the degree 3 surfaces, and one may recover the Perdomo’s result essentially by localanalysis. 4.
Degree 3 algebraic minimal surfaces
In this section, we give a local analytic characterization of degree 3 algebraic minimal surfaces in S . Theorem 4.1. [Pe2]
Let x :
M ֒ → S ⊂ E be an algebraic minimal surface of degree 3. Then thestructure functions satisfy h h − + h − h = 0 ,h h − + 4 h h − + 4 h h − = 10 ( h h − ) . For an orthonormal coordinate { x , x , x , x } of E , the surface is defined by the cubic polynomial F = − x x x + x ( x − x ) . a) x is the conjugate surface of a principally bi-planar minimal surface. It has the Killing nullity 1,and there exists up to scale a unique Killing vector field of SO that is tangent to x .b) The curvature of the minimal surface takes values in the closed interval [ − , ] . This does not necessarily mean that the higher degree algebraic minimal surfaces are characterized by a set of higherorder equations. But the differential analysis itself does require manipulation of the higher order terms.
EGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE 11
The minimal surface defined by the above cubic polynomial is one of the infinite sequence of algebraicminimal tori constructed by Lawson, [La, p350].
Remark 4.2.
Given the degree 3 algebraic minimal torus, one may ask if the conjugate surface, whichis principally bi-planar, is also algebraic. An analysis indicates that this conjugate minimal surface doesnot close up to become a torus.
Perdomo first gave the characterization of degree 3 algebraic minimal surfaces in S , [Pe2]. One ofthe main idea of his analysis is that such a minimal surface necessarily contains a great circle, andthat the gradient of the defining cubic polynomial vanishes at a point on the great circle. This putsthe cubic polynomial into a special normal form. By applying the minimal surface equation (1.2),the characterization is reduced essentially to solving a set of algebraic equations among the constantpolynomial coefficients.The analysis for the degree 3 case proceeds similarly as for the degree 1, and the degree 2 cases treatedin the previous section. On the other hand, a cubic polynomial on E = R has 20 coefficients. Thecomputational complexity increases as degree increases, and one has to take higher order derivatives inorder to access the compatibility equations for a minimal surface to be algebraic. For the degree 3 case,the analysis requires differentiating six times.Proof of Theorem 4.1 consists of two parts. In the first part, a preliminary analysis is done to reducethe number of independent polynomial coefficients from 20 to 6, Section 4.1. After differentiating fourtimes, the analysis divides into two cases depending on whether a certain structure invariant of theminimal surface vanishes or not. In the second part, we carry out the differential analysis for each caseunder the appropriate assumptions on the structure functions, Section 4.2, Section 4.3.For the rest of the paper, we assume h ± = 0 , − h h − = 0, and h ± = 0 to exclude degree 1, anddegree 2 algebraic minimal surfaces.4.1. Preliminary analysis.
Let us write a real, homogeneous degree 3 polynomial F ∈ S ( E ) in termsof the frame e C , (2.7). F = p e + p e e + p e e + p e (4.3) + 8 q E + 8 q E E − + 8 q E E − + 8 q E − + 2 r , e E + 2 r , − e E − + 2 r , e e E + 2 r , − e e E − + 2 r , e E + 2 r , − e E − + 4 r , e E + 4 r , e E E − + 4 r , − e E − + 4 r , e E + 4 r , e E E − + 4 r , − e E − . We implicitly assume the appropriate conjugation relations among the coefficients so that F is real, i.e., r , − = r , , etc. F ∈ S ( E ) is a constant cubic polynomial. Differentiating (4.3) by Leibniz rule, the equation dF = 0is equivalent to the following set of structure equations for the polynomial coefficients. dp − r , − ω − r , ω = 0 , (4.4) dp + ( − r , − + h r , ) ω + ( − r , + h − r , − ) ω = 0 ,dp + ( − r , − + h r , ) ω + ( − r , + h − r , − ) ω = 0 ,dp + h r , ω + h − r , − ω = 0 ,dq − q ρ + 12 r , ω − h − r , ω = 0 ,dq − i q ρ + 12 ( r , − h r , ) ω + 12 ( r , − h − r , ) ω = 0 ,dq + i q ρ + 12 ( r , − − h r , ) ω + 12 ( r , − h − r , − ) ω = 0 ,dq + 3 i q ρ − h r , − ω + 12 r , − ω = 0 ,dr , − i r , ρ + 12 ( − r , + 3 p ) ω + 12 ( − r , − h − p ) ω = 0 ,dr , − + i r , − ρ + 12 ( − r , − − h p ) ω + 12 ( − r , + 3 p ) ω = 0 ,dr , − i r , ρ + ( − r , + p + 2 h r , ) ω + ( − r , − h − p + h − r , ) ω = 0 ,dr , − + i r , − ρ + ( − r , − − h p + h r , ) ω + ( − r , + p + 2 h − r , − ) ω = 0 ,dr , − i r , ρ + 12 ( p + 4 h r , ) ω + 12 (2 h − r , − h − p ) ω = 0 ,dr , − + i r , − ρ + 12 (2 h r , − h p ) ω + 12 ( p + 4 h − r , − ) ω = 0 ,dr , − r , ρ + ( − q + r , ) ω + 12 ( − q − h − r , ) ω = 0 ,dr , + 12 ( − q − h r , + 2 r , − ) ω + 12 ( − q − h − r , − + 2 r , ) ω = 0 ,dr , − + 2 i r , − ρ + 12 ( − q − h r , − ) ω + ( − q + r , − ) ω = 0 ,dr , − r , ρ + 12 ( r , + 6 h q ) ω + ( − h − r , + h − q ) ω = 0 ,dr , + 12 ( r , − + 4 h q − h r , ) ω + 12 ( r , + 4 h − q − h − r , − ) ω = 0 ,dr , − + 2 i r , − ρ + ( − h r , − + h q ) ω + 12 ( r , − + 6 h − q ) ω = 0 . Differentiating the coefficients of F would mean applying this structure equation from now on.The preliminary analysis consists of the following 5 steps. The condition that F vanishes on theminimal surface serves as the initial condition for the over-determined PDE analysis. By successivelyapplying the above structure equation, we reduce the number of independent polynomial coefficients to20 −
14 = 6. The reduction process stops at
Step 5 , where the analysis divides into two cases.
Step 1 . Assume the initial condition Eq , : p = 0 . By the metric duality, this equivalent to that the cubic polynomial F vanishes on the minimal surface x. EGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE 13
Step 2 . Differentiating p = 0, one gets Eq , : r , = 0 ,Eq , : r , − = 0 . Step 3 . Differentiating r , ± = 0, one gets Eq , : 12 h − p + 2 r , = 0 , (4.5) Eq , : r , = 0 ,Eq , : 12 h p + 2 r , − = 0 . One may solve for { r , , r , , r , − } from these equations. Step 4 . Differentiating (4.5), one gets Eq , : 2 q + 12 h r , = 0 , (4.6) Eq , : 2 q + 12 h − r , − = 0 ,Eq , : 6 q + 12 p h − + 32 h − r , = 0 ,Eq , : 6 q + 12 p h + 32 h r , − = 0 . One may solve for { q , q , q , q } from these equations. Step 5 . Differentiating (4.6), one gets Eq , : 2 h r , + 6 h r , + ( − h + h h − ) p = 0 , (4.7) Eq , : h − r , − + h r , + 12 h h − p = 0 ,Eq , : 2 h − r , − + 6 h − r , + ( − h − + h − h ) p = 0 ,Eq , : 12 h − h r , + 6 h h − r , − − h − h h r , + ( − h h h − − h − h + 2 h − h h − ) p = 0 ,Eq , : 12 (cid:0) h h − + h − h (cid:1) r , + ( − h h − h − h − h h − + 2 h h − h − h − h h h − − h − h + 2 h − h h − ) p = 0 . Since we are assuming that the algebraic minimal surface has degree 3, both h ± , h ± do not vanishidentically, otherwise the algebraic minimal surface would have degree 1, or 2 by Theorem 3.6, Theorem3.8. One may thus solve for { r , , p , r , − , r , } from { Eq , , Eq , , Eq , , Eq , } . Note that Eq , isequivalent to Eq , modulo Eq , . (cid:3) At this step, the analysis is divided into the following two cases. Set∆ +3 = h − h + h h − , (4.8) ∆ = h h − h − h − h h − . • Case ∆ +3 = 0, Section 4.2: It will be shown that the degree 3 algebraic minimal surface in Theorem4.1 belongs to this case. The minimal surfaces with ∆ +3 = 0 are the conjugate surfaces of the principallybi-planar minimal surfaces. A minimal surface in S is principally bi-planar when each of its principalcurves is planar, [Ya]. The principally bi-planar minimal surfaces are characterized by the equation ∆ − = h − h − h h − = 0. Both of the ∆ ± -null minimal surfaces belong to the wider class of thecohomogeneity 1 minimal surfaces, [HsL, p32]. • Case ∆ +3 , ∆ = 0 , Section 4.3: The analysis shows that for a minimal surface with ∆ +3
0, thefourth order equation ∆ = 0 is a necessary condition to be algebraic of degree 3. But, a further analysisshows that the resulting structure equation is not compatible, and there does not exist any degree 3algebraic minimal surfaces in this case.In the following two sections, we present the differential analysis for each case.4.2. Case h − h + h h − = 0 . In this section, we show that, up to motion by SO , there exists a uniquedegree 3 algebraic minimal surface in S that satisfies ∆ +3 = h − h + h h − = 0.We first determine the structure equation for the minimal surfaces with ∆ +3 = 0, Section 4.2.1. A firstintegral J is defined, (4.10), and it follows that there exists locally a one parameter family of ∆ +3 -nullsurfaces. We then continue the analysis of Section 4.1 and show that exactly one of the ∆ +3 -null surfacesis algebraic of degree 3, Section 4.2.2.4.2.1. Structure equation.
Let x :
M ֒ → S be a ∆ +3 -null surface. Differentiating ∆ +3 = 0, one may solvefor h ± and get h = 12 h h − ( − h h − + 2 h h − − h h − ) h h − , (4.9) h − = h . Here we assume h ± , h ± are nonzero. A direct computation shows that the structure equation iscompatible with this relation, i.e., d h ± = 0 is an identity.Set(4.10) J = 18 (cid:0) h h − + 4 h h − + 4 h h − (cid:1) ( h h − ) , J > . Then d J = 0, and J is a first integral for the ∆ +3 -null surfaces.We claim that J ∈ (1 , ∞ ), and that h is nowhere zero. Let us denote | h | = a, | h | = b . Theequation (4.10) is written as(4.11) b + 4 a (1 + a − a ) = 0 . This implies that J ≥
1, and that either a ≡
0, which is excluded, or a takes values in the closed interval[J − p J − , J + p J − a ≡ , b ≡
0, which is also excluded.The structure equations (4.9) and (4.10) will be used implicitly for the analysis in the next subsection.4.2.2.
Differential analysis.
We now continue the analysis of Section 4.1. Due to their lengths, the exactexpressions for Eq , , Eq , , Eq , , Eq , ; Eq below will be postponed to Appendix. Step 5’ . Assume ∆ +3 = 0. Then (4.9) implies that Eq , = 0. Step 6 . Differentiating (4.7) and equating modulo ∆ +3 = 0, one gets a set of four independentequations { Eq , , Eq , , Eq , , Eq , } (see Appendix). One may solve for { r , , r , , r , − r , − } fromthese equations. Step 7 . Differentiating Eq , , one gets Eq (see Appendix). One may solve for p from this equation.At this step, p is the only remaining independent polynomial coefficient, and it satisfies the structureequation, (4.4), of the form dp ≡ , mod p . One may verify this by a straightforward moving frame computation. We omit the details. One may verify by direct computation that the cohomogeneity 1 minimal surfaces in S are characterized by the pairof fourth order equations ∆ +4 = 2 h h − h − h h − (3 h h − + 2 h h − K ) = 0 , ∆ − = ∆ +4 = 0. EGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE 15
From the uniqueness theorem of ODE, if p vanishes at a point of the minimal surface, it vanishesidentically, which implies the cubic polynomial F vanishes. We therefore assume p is nowhere zerofrom now on. Step 8 . Differentiating the remaining equations { Eq , , Eq , , Eq , ; Eq } , one gets a single compat-ibility equation, up to scale by non-identically zero terms;(8J + 10)(8J −
10) = 0 . Here J is the first integral (4.10). Since J >
1, one must have(4.12) J = 54 . Moreover, a direct computation shows that d p = 0 is an identity with this relation.By the existence and uniqueness theorem of ODE, there exists up to scale a unique nonzero cubicpolynomial that vanishes on the ∆ +3 -null minimal surface with the first integral J = .Fix a point x on the minimal surface. Choose an orthonormal coordinate { x , x , x , x } of E sothat one has the following identification at x by the metric duality. e = x ,E = 12 ( x − i x ) exp (i π ,E − = 12 ( x + i x ) exp ( − i π ,e = x . Up to scale, one may assume p = at x . From the analysis of Section 4.2.1, | h | = a takes values inthe closed interval [ , a → F with h = a , the degree 3 polynomial F is given by(4.13) F = − x x x + x ( x − x ) . Proof of Theorem 4.1. a) From (4.13), set z = x + i x , z = x + i x . Then F = Re ( z z ). It is clear that F is invariantunder a subgroup SO ⊂ SO . It is known that a minimal surface in S with Killing nullity ≥ | h | = a takes values in the closed interval [ , K = 1 − a . (cid:3) Case h − h + h h − = 0 . In this section, we show that there does not exist a degree 3 algebraicminimal surface in S with the property that ∆ +3 = h − h + h h − is not identically zero.The analysis will show that under the condition ∆ +3
0, a degree 3 algebraic minimal surface neces-sarily satisfies a fourth order equation ∆ = h h − h − h − h h − = 0. We first determine the structureequation for the minimal surfaces with ∆ = 0, Section 4.3.1. We then continue the analysis of Section4.1 and show that none of the ∆ -null surfaces is algebraic of degree 3, Section 4.3.2.4.3.1. Structure equation.
Let x :
M ֒ → S be a ∆ -null surface. Differentiating ∆ = 0, one may solvefor h ± and get h = 5 h − h h − − h − h h − h + 4 h h − h h − − h h − h h − + 4 h h − h h − − h h − h h − h , (4.14) h − = h . Here we assume h ± , h ± are nonzero. A direct computation shows that the structure equation iscompatible with this relation, i.e., d h ± = 0 is an identity.The structure equation (4.14) will be used implicitly for the analysis in the next subsection. Differential analysis.
We now continue the analysis of Section 4.1. Due to their lengths, the exactexpressions for Eq , , Eq , , Eq , ; Eq below will be postponed to Appendix. Step 5’ . Assume ∆ +3 = 0. One may solve for r , from Eq , . Step 6 . Differentiating Eq , , Eq , from (4.7), one gets a set of three independent equations { Eq , , Eq , , Eq , } (see Appendix). One may solve for { r , , r , − , r , − } from these equations. Step 7 . Differentiating Eq , , one gets a single equation Eq (see Appendix). One may solve for p from this equation.At this step, p is the only remaining independent polynomial coefficient, and it satisfies the structureequation, (4.4), of the form dp ≡ , mod p . As in Section 4.2.2, we assume p is nowhere zero from now on. Step 8 . Differentiating Eq , , Eq , from (4.7), one gets a set of two equations, which allow one tosolve for h ± (see Appendix). Note that we have not assumed ∆ = 0 yet.At this step, we observe that the coefficient p is real. Evaluating p − p = 0 with the relationsobtained so far, one gets the following compatibility equation;∆ ∆ ′ = 0 , where ∆ ′ = ( − h h − h − + 3 h h − h − + 10 h h − − h h − h − + 3 h h h − − h h − h − h h − h + 10 h − h ) . A short analysis shows that ∆ ′ vanishes only when h ≡
0, which is excluded. Hence a degree 3 algebraicminimal surface with ∆ +3 = 0.We assume the structure equations (4.14) from now on. Step 9 . The cubic polynomial F , (4.3), is real. Evaluating F − F = 0, one gets the single compatibilityequation ( − h − h + h h − )∆ ′′ = 0 , where ∆ ′′ = 3 h h − h + (3 h h h − − h h h − − h − h ) h − h − h h + 10 h − h . A further differential analysis shows that ( − h − h + h h − ) vanishes only when h ≡
0, which isexcluded (the differential analysis for this case is a little evolved, but straightforward. We shall omit thedetails). Hence a degree 3 algebraic minimal surface with ∆ +3 ′′ = 0. Step 10 . Differentiating Eq , modulo ∆ ′′ = 0, one gets another single compatibility equation, up toscale by non-identically zero terms; Eq , : (3 h h − − h − h h − = 0 . Comparing Eq , with ∆ ′′ = 0, one gets Eq , : 39 h h − − h h − + 20 + 16 h h − = 0 . Differentiating this equation again, one gets Eq , : 93 h h − − h h − + 40 + 32 h h − = 0 .Eq , and Eq , are compatible only when h ± = 0. (cid:3) EGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE 17
Appendix.
We record the exact formulae of the long expressions omitted in the main text.A-1.
Section 4.2 Eq , : ( − h + 6 h h − ) r , − h h − h r , − + 4 h h − r , − + ( h h − − h h − h + h − h h − ) p = 0 ,Eq , : (36 h h − h − h − h h − h − h ) r , + (54 h − h h − h + 72 h h − h − h h − h + 36 h − h h − + 48 h − h h h − − h − h h − ) r , − + ( − h − h h − h − h h − h − + 24 h h − h − h − h h h − + 48 h h − h − h h − h h − ) r , − + ( − h h − h − + 8 h h − h h − + 6 h − h h − + 10 h h − h − h + 9 h h − h − h − h h − h − h h − h − h ) p = 0 ,Eq , : ( − h h − h + 102 h − h h − h − h − h h − + 36 h h − h + 36 h − h h − + 48 h − h h h − t ) r , − + (24 h h − h − h h h − − h h − h h − + 48 h h − h − h − h − h h − h − h h − h − ) r , − + (14 h h − h h − − h h h − h − − h h − h − + 6 h h − h − h + 17 h h − h − h − h h − h + 6 h − h h − ) p = 0 ,Eq , : ( − h − h h h − + 108 h − h h − h ) r , − + ( − h h − h − + 42 h h − h − + 34 h h − h − h − h h − h − − h h − h − h − h h h − + 24 h − h h − h + 34 h h − h h − − h h − h h − + 18 h h − h − h − h h − h h − − h − h h − h h h − h − − h h − h − + 18 h h − h + 9 h − h h − h ) p = 0 ,Eq : ( − h h − h − h + 162 h h − h h − ) p + (108 h h − h h − + 894 h h − h − h + 294 h h − h h − − h h h − h − + 452 h h h − h − − h h − h − h + 216 h h − h − h − h h − h h − + 306 h h − h h − − h h − h − h − h − h h h − + 72 h h − h − h + 114 h h − h h − − h h − h h − + 1092 h h − h h − − h h − h h − + 72 h h − h − h − h h h − h − + 108 h h − h h − − h − h h h − − h h h − h − + 144 h − h h − h − h h − h h − + 374 h h − h h − − h h − h − h + 144 h − h h + 432 h h − h − − h h − h − h h − h + 72 h − h − h − h h − h − − h − h h − − h h − h − ) p = 0 . A-2.
Section 4.3 Eq , : (36 h h − + 36 h h h − ) r , + ( − h − h + 6 h h h − h − + 6 h h h h − − h h h h − − h h − h h − − h h h − h − + 10 h h h − h − + 4 h h − h h − + 4 h h − h h − − h h − h − h − h h h − + 10 h h − h ) p = 0 ,Eq , : ( − h h h − h − − h h − ) r , − + (12 h h − h − + 12 h h − h ) r , − + ( h h − h − h + 2 h h − h h − − h h − h − h − − h h − h − h + h h − h − + 2 h h − ) p = 0 Eq , : (36 h h − h − h + 36 h h − h − ) r , − + (6 h − h − h h − h h h − + 3 h − h h h − h − h h h − + 4 h − h h h − + 3 h − h h h − + 10 h h h − h − − h − h h h − − h h − + 4 h h h − h h − + 4 h h − h − h − h h − h − h h − h − h h − h − h − h h − h − + 3 h h − h − + 3 h − h h h − − h − h h h ) p = 0 ,Eq : (216 h − h h − + 432 h h − h h − + 216 h h − h ) p + (20 h h − h + 40 h h h − − h h − h h − h h − + 18 h h h − h h − + 18 h h − h h − h − − h h − h h h − − h h − h h − h − h − h h h − h − − h h − h − h h − + 12 h h − h h − h h − + 32 h h − h h h − + 8 h h − h h − h − h h − h h − h − + 44 h h h − h − h − + 54 h h − h h − h − h − h h − h h + 36 h h h − h h − h − − h h h − h h − − h h − h h − + 24 h h − h h − + 38 h h h − h h − + 24 h h − h h − h − + 54 h h h − h − h − + 16 h h − h h − − h h − h h − − h − h h h − − h h − h h − − h h − h h − h h − − h h − h h − h − h h − h h − h − + 24 h − h h − h − + 16 h h h − h − + 15 h h − h − h + 40 h h − h h − + 16 h h h − h − + 36 h h h − h − + 24 h h − h h − − h − h h − h − − h h − h h − − h h − h h − h − + 48 h h − h h − h − h h h − − h h − h h − + 48 h h − h h − − h h − h h + 52 h h − h h − − h h − h h − + 24 h − h h h − − h h − h h − + 16 h h − h h − h − − h − h h h + 3 h h − h h − + 24 h − h h − h − + 24 h − h h h − − h h h − h − − h − h h h − − h − h − h h − h − + 40 h h − h − + 56 h − h h − + 68 h h − h − h h − h ) p = 0 . EGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE 19
Step 8 in Section 4.3.2, formulae for h ± : h = − h h − h − ( h h − + h − h ) (36 h − h h − h − h + 12 h h − h h − h − h h − − h h − h − h h − h h h − h − h − − h h − − h h − h h h − + 20 h h − h h − − h h h − h − − h h − h h − − h h − h − h + 4 h h − h h − + 15 h h − h − h − + 3 h h h − h − + 3 h h h − h − h h − h h − + 15 h h − h h − − h h h − h − h h h − h − − h h − h − + 4 h h h − + 12 h h − h h − h − + 20 h h − h h h − − h h − h h h − − h h − h − h h − h h − h − h − h h + 3 h h − h h − h − ) ,h − = 112 h h h − ( h h − + h − h ) ( − h h − h h + 30 h − h h h − h − h + 12 h h − h − h + 50 h − h h h − − h h − h h − − h h − h − h h + 10 h h − − h h − h − h − h h − h h − + 3 h h − h h h − + 10 h − h h − h + 27 h − h h h − h − + 40 h h − + 12 h h − h h − − h h − h − h − − h − h h h − + 12 h h − h h + 12 h h − h − h − − h − h h − + 26 h h − h + 20 h − h h h − h − ) . Note that h − = h in this formula. h − − h = 0 gives an integrability equation which is quadratic in h , h − . References [Ha] Haskins, Mark,
Special Lagrangian cones , Am. J. Math. 126, No. 4 (2004), 845-871[Hs] Hsiang, Wu-yi,
Remarks on closed minimal submanifolds in the standard Riemannian m -sphere , J. Differential Geom-etry, 1 (1967), 257–267[HsL] Hsiang, Wu-yi; Lawson, B., Minimal submanifolds of low cohomogeneity , J. Differential Geometry, 5 (1971), 1–38[La] Lawson, B.,
Complete minimal surfaces in S Ann. Math. (2) 92 (1970), 335–374[Pe1] Perdomo, Oscar Mario,
Non-existence of regular algebraic minimal surfaces of spheres of degree 3 , J. Geom. 84, No.1-2 (2005), 100–105[Pe2] Perdomo, Oscar Mario,
Characterization of order 3 algebraic immersed minimal surfaces of S , Geom. Dedicata 129(2007), 23–34[Ya] Yamada, Kotaro, Minimal tori in S whose lines of curvature lie in S , Tokyo J. Math. 10 (1987), 215–226 E-mail address ::