Minimal Surface System in Euclidean Four-Space 3

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Theorem 1.2 (Two applications of Lagrange potential of height on minimal surfaces). Let Σ be the minimal graph of the function p : Ω R deﬁned on a domain Ω R2 : 0 → ⊂ Σ x, y,p(x, y) R3 (x, y) Ω , 0 = ∈ | ∈ where the height function p(x, y©¡) solves the minimal¢ surfaceª equation

p ∂ px ∂ y Ω 0    , x, y . = ∂x 2 2 + ∂y 2 2 ∈ 1 px py 1 px py  + +   + +  ¡ ¢ q q Let q : Ω R denote the Lagrange potential of p : Ω R, which solves the system → →

py px (1.1) qx,qy  , . = − 2 2 2 2 1 px py 1 px py ¡ ¢  + + + +   q q  (a) For any constant λ R {0}, we associate the two dimensional graph in R4 ∈ − (1.2) Σ x, y,(coshλ) p(x, y),(sinhλ) q(x, y) R4 (x, y) Ω . λ = ∈ | ∈ Then, the pair©¡ f (x, y), g(x, y) (coshλ)p(x, y),¢(sinhλ)q(x, y) ªobeys the Osser- = man system with the coefﬁcient µ cothλ. In particular, the graph Σ becomes a ¡ ¢ ¡= ¢ λ minimal surface in R4. Furthermore, we obtain the invariance of the conformally 1 1 changed induced metric gΣλ gΣ0 . Σ Σ = det g 0 det g λ r q (b) For any constant λ R {0}, we³ associate´ the¡ three¢ dimensional graph in R6 ∈ − p p y x R6 Ω R x, y, z, px λz , py λz , λq (x, y) , z . − 2 2 + 2 2 ∈ | ∈ ∈  1 px py 1 px py   + + + +   q 3 q  6  Then, it is special Lagrangian in C , so homologically volume minimizing in R. MINIMAL SURFACE SYSTEM IN EUCLIDEAN FOUR-SPACE 3

We ﬁnd the one parameter family of two dimensional minimal graphs in R4 deﬁned over punctured plane, which realizes (part of) the Lagrangian catenoid in C2 and Fraser- Schoen band in R4 (Example 2.3). We have the two parameter family of minimal graphs in R4 connecting catenoids, helicoids, planes in R3, and complex logarithmic graph in C2 (Example 2.5). Applying the item (a) to helicoid and catenoid in R3, we obtain minimal surfaces in R4 foliated by hyperbolas or lines (Example 4.4) and the Osserman- Hoffman ellipse-foliated minimal annuli in R4 with total curvature 4π (Example 4.5) − respectively. Applying the item (b) to Scherk’s graph R3, we build the doubly periodic special Lagrangian graphs in C3 (Example 5.5).

2. MINIMAL SURFACE SYSTEM IN R4 AND CAUCHY–RIEMANN EQUATIONS Our ambient space is Euclidean space R4 equipped with the metric d x 2 d x 2. 1 +···+ 4 Theorem 2.1 (Two dimensional minimal graphs in R4). Let Σ denote the graph in R4 Σ2 Φ(x, y) x, y, f (x, y), g(x, y) R4 (x, y) Ω . = = ∈ | ∈ The induced metric gΣ on© the surface¡ Σ reads ¢ ª 2 2 gΣ Ed x 2Fdxdy Gd y , = + + where the coefﬁcients of the ﬁrst fundamental form are determined by (2.1) E Φ Φ 1 f 2 g 2, F Φ Φ f f g g , G Φ Φ 1 f 2 g 2. = x · x = + x + x = x · y = x y + x y = y · y = + y + y The areaelementis dAΣ ωdxdy wtih ω pEG F 2. We introduce the minimal surface = = − operator LΣ and Laplace-Beltrami operator Σ acting on functions on Ω : △ ∂2 ∂2 ∂2 (2.2) L : G 2F E , Σ = ∂x2 − ∂x∂y + ∂y2

1 ∂ G ∂ F ∂ ∂ F ∂ E ∂ (2.3) Σ : gΣ . △ = △ = ω ∂x ω ∂x − ω ∂y + ∂y −ω ∂x + ω ∂y · µ ¶ µ ¶ ¸ Then, the following three conditions are equivalent. (a) Two height functions f (x, y) and g(x, y) are harmonic on the graph Σ: f 0 and g 0. △Σ = △Σ = (b) The graph Σ is minimal in R4. (c) Two height functions f (x, y) and g(x, y) solve the minimal surface system: (2.4) L f 0 and L g 0. Σ = Σ = Proof. The equivalence of (a) and (b) follows from [34, Equation (3.14) in Section 2], which indicates that the Euler-Lagrange system for the area functional of the graph is

∂ G fx F fy ∂ F fx E fy 0 fx 0 Σ . ∂x ω gx − ω gy + ∂y −ω gx + ω gy = 0 ⇔ △ gx = 0 µ · ¸ · ¸ ¶ µ · ¸ · ¸ ¶ · ¸ · ¸ · ¸ There are several ways to establish the equivalence of (b) and (c):[34, Section 2, p. 16-17], [24, Section 2], [2, Section 1.2] (for arbitrary codimension), [29, Appendix: The minimal 4 HOJOO LEE surface system], and [10, Example 1] (for more general ambient spaces). Here, we adopt the argument in the proof of [32, Theorem 2.2]. We use the formula (2.3) and introduce ∂ G ∂ F ∂ E ∂ F (2.5) (P ,Q) , . = ∂x ω − ∂y ω ∂y ω − ∂x ω µ µ ¶ µ ¶ µ ¶ µ ¶¶ to obtain the identity for the mean curvature vector H(x, y) : Φ(x, y) explicitly: = △Σ x P 0 y 1 Q P Q 1 0 (2.6) H Φ Φ . Σ   P Q 1 L  x y 2 L  = △ f (x, y) = ω fx fy Σ f = ω + ω + ω Σ f + + ω g(x, y) P g Qg 1 L g L g     x y ω Σ   Σ     + +    First, assume (b). Since the mean curvature vector H vanishes on the minimal surface, we ﬁnd that (2.6) guarantees that four quantities P , Q, LΣ f , LΣ g vanish. So, (c) holds.

Second, assume (c). Since LΣ f 0 and LΣ g 0, we ﬁnd that, by (2.6), the mean curva- = P = Q ture vector H is equal to the tangent vector Φ Φ . As the mean curvature vector H ω x + ω y is normal to the graph Σ, we conclude that H vanishes. So, (b) holds.

Remark 2.2 (Minimal surface operator LΣ and Laplace-Beltrami operator Σ). When △ we assume that the two dimensional minimal graph Σ is minimal in R4, we obtain G ∂2 F ∂2 E ∂2 1 Σ 2 , or brieﬂy, Σ LΣ. △ = ω2 ∂x2 − ω2 ∂x∂y + ω2 ∂y2 △ = ω2 The minimality of the graph Σ implies the two interesting identities ∂ F ∂ G ∂ E ∂ F (2.7) and , ∂y ω = ∂x ω ∂y ω = ∂x ω µ ¶ µ ¶ µ ¶ µ ¶ which imply 1 ∂ G ∂ F ∂ ∂ E ∂ F ∂ 1 Σ LΣ LΣ. △ = ω2 + ∂x ω − ∂y ω ∂x + ∂y ω − ∂x ω ∂y = ω2 · µ ¶ µ ¶¸ · µ ¶ µ ¶¸ A geometric implication of (2.7) on minimal graphs can be found in Rado’s book [36, p. 108]. A variational proof of (2.7) is given in Osserman’s book [34, Chapter 3]. Another interpretation of (2.7) (via conjugate minimal surface) is illustrated in Remark 4.2.

By Bers’ Theorem [3, 11, 33], an isolated singularity of a single valued solution to the minimal surface equation is removable. However, in higher codimensions, the minimal surface system can have solutions with isolated non-removable singularities.

Example 2.3 (Two dimensional minimal graphs in R4 over the punctured plane). Let N be a positive integer. We shall modify the entire holomorphic graph in C2 R4 = Σ x, y,Re x i y N ,Im x i y N R4 (x, y) R2 . entire = + + ∈ | ∈ n ³ h¡ ¢ i h¡ ¢ i´ o Let TN (ζ) denote the Chebyshev polynomial of the ﬁrst kind and degree N. The Cheby- shev polynomials are determined by the recurrence relation 2 T0 (ζ) 1, T1 (ζ) ζ, T2 (ζ) 2ζ 1, Tk 2 (ζ) 2ζTk 1 (ζ) Tk (ζ) for k 0. = = = − + = + − ≥ MINIMAL SURFACE SYSTEM IN EUCLIDEAN FOUR-SPACE 5

The Chebyshev polynomial TN (ζ) is characterized by the properties

cosh(N arcoshζ) for ζ [1, ], TN (ζ) ∈ ∞ = (cos(N arccos ζ) for ζ [ 1,1]. ∈ − We construct the following minimal graph over the punctured plane

N N Σ x, y,Ψ (ρ)Re x i y ,Ψ (ρ)Im x i y R4 (x, y) R2 {(0,0)} , N = N + N + ∈ | ∈ − n ³ h i h i´ o where we introduce the radially¡ ¢ symmetric weight¡ function¢ ΨN (ρ):

2 2 2 1 TN 1 ρ 1 TN 1 x y ρ x2 y2 and Ψ (ρ) + + + . = + N = · ³pN ´ = · ³p N ´ N ρ N x2 y2 2 q + (a) The new coordinates (t,θ) (0, ) [0,2π] with x, y (sinh¡ t cos¢ θ,sinh t sinθ) ∈ ∞ × = induce the conformal harmonic patch which admits the extension X (t,θ): ¡ ¢ N 1 1 (t,θ) R [0,2π] sinh t cosθ,sinh t sinθ, cosh(Nt)cos(Nθ), cosh(Nt)sin(Nθ) ∈ × → N N µ ¶ 4 of the minimal surface ΣN in R with the induced conformal metric 2 2 2 2 2 2 2 2 gΣ cosh t sinh (Nt) dt dθ sinh t cosh (Nt) dt dθ . N = + f + = + + (b) When£ N 1, we have the¤¡ minimal gradient¢ £ graph Σ explicitly¤¡ given by ¢ f = 1 1 1 x, y, x 1 , y 1 R4 (x, y) R2 {(0,0)} , s + x2 y2 s + x2 y2 ∈ | ∈ − ( Ã + + ! ) As Osserman [33] observed, we see that the pair of height functions

x y f (x, y), g(x, y) 1 x2 y2 , = + + Ã x2 y2 x2 y2 ! ¡ ¢ q + + has an isolated singularity at the origin.p Neither heightp functions f (x, y) nor g(x, y) tends to a limit at the origin. The Lagrangian catenoid ([15, Theorem 3.5] and [26, Example 6.4]) realizes the surface given by the patch X1(t,θ). (c) When N 2, the minimal graph Σ becomes a rational variety = 2 1 1 x, y, x2 y2 1 , 2x y 1 R4 (x, y) R2 {(0,0)} . − + 2 2 + 2 2 ∈ | ∈ − ½ µ µ 2 x y ¶ µ 2 x y ¶ ¶ ¾ ¡ ¢ + ¡ ¢ + ¡ ¢ ¡ ¢ 1 It is the graph of the non-holomorphic function ζ x i y C {0} ζ2 1 ζ − ζ. = + ∈ − 7→ + 2 The Fraser-Schoen band [13, Section 7] is the minimal embedding X2(t,³θ)ofa´ Möbius band into R4. This induces, after a suitable rescaling, the free boundary minimal surface in the four dimensional unit ball whose coordinate functions become ﬁrst Steklov eigenfunctions [13, Proposition 7.1].

Remark 2.4. Osserman [33] extended Bers’ Theorem to the minimal surface system in arbitrary codimensions, assuming the continuity of all but one of the height functions. 6 HOJOO LEE

Example 2.5 (Two parameter family of minimal graphs in R4 connecting catenoids, helicoids in R3, and complex logarithmic graph in C2). Given a pair (α,β) R2 of con- Σ R4 ∈ stants, we associate the following two dimensional graph (α,β) in x2 y2 x2 y2 β2 α2 Σ y R4 Ω (α,β) x, y,αln + + + + − ,βarctan (x, y) . = (Ã Ãp p 2 ! x ! ∈ | ∈ ) ³ ´ r pr 2 β2 α2 y The pair f (x, y), g(x, y) αln + + − ,βarctan with r x2 y2 solves = 2 x = + µ µ ¶ ¶ the minimal¡ surface system¢ (2.4) in Theorem 2.4. We shall¡ distinguish¢ thep three cases: (a) When (α,β) (coshλ,sinhλ) for some constant λ R, recall the identity = ∈ arcoshr ln r r 2 1 , r 1. = + − ≥ ³ p Σ ´ We see that, up to translations, the graph (α,β) is congruent to

2 2 y 4 2 2 Σ+ x, y,(coshλ) arcosh x y ,(sinhλ) arctan R x y 1, x 0 . λ = + x ∈ | + ≥ 6= ½µ µq ¶ ³ ´¶ ¾ When λ 0, the surface Σ+ recovers the catenoid foliated by circles. = 0 (b) When (α,β) (λ,λ) for some λ R {0}, the non-planar minimal surface Σ = ∈ − (α,β) in R4 can be identiﬁed as the complex logarithmic graph in C2: Σ0 ζ,λlogζ C2 ζ C {0} . λ = ∈ | ∈ − (c) When (α,β) (sinhλ,cosh©¡ λ) for some¢ constant λ ªR, recall the identity = ∈ arsinhr ln r r 2 1 , r R. = + + ∈ ³Σ p ´ We ﬁnd that, up to translations, (α,β) is congruent to

2 2 y 4 Σ− x, y,(sinhλ) arsinh x y ,(coshλ) arctan R (x, y) (R {0}) R . λ = + x ∈ | ∈ − × ½µ µq ¶ ³ ´¶ ¾ In the case when λ 0, the surface Σ− recovers the helicoid foliated by lines. = 0 Proposition 2.6 (Cauchy–Riemann equations on the two dimensional minimal graph). Assume that the graph Σ x, y, f (x, y), g(x, y) R4 (x, y) Ω is minimal in R4. Then, = ∈ | ∈ the C-valued function A (x, y) iB(x, y) is holomorphic on Σ when we have ©¡ + ¢ ª A E F B B E F A (2.8) x ω ω y , or equivalently, x ω ω y . A = F G B B = − F G A · y ¸ · ω ω ¸·− x ¸ · y ¸ · ω ω ¸·− x ¸ Proof. We use the special construction of the local conformal coordinates on minimal graphs to deduce the Cauchy–Riemann equations (2.8) on Σ. Due to the identities (2.7): ∂ F ∂ G ∂ E ∂ F and , ∂y ω = ∂x ω ∂y ω = ∂x ω µ ¶ µ ¶ µ ¶ µ ¶ we can ﬁnd the potential functions M(x, y) and N(x, y) so that F G E F Mx ,My , and Nx,Ny , , = ω ω = ω ω µ ¶ µ ¶ ¡ ¢ ¡ ¢ MINIMAL SURFACE SYSTEM IN EUCLIDEAN FOUR-SPACE 7 at least in a sufﬁciently small neighborhood of any point in Ω. As in [34, Lemma 4.4],

x, y Ξ(x, y) (ξ ,ξ ) x M(x, y), y N(x, y) → = 1 2 = + + becomes the local diffeomorphism,¡ ¢ and gives¡ the desired local conformal¢ coordininates 4 (ξ1,ξ2) on the minimal graph Σ in R equipped with the induced conformal metric

ω 2 2 gΣ dξ dξ . = E G 1 + 2 2 ω ω + + ¡ ¢ The C-valued function A (x, y) iB(x, y) is holomorphic on Σ with respect to the confor- + mal coordinates (ξ ,ξ ) Ξ(x, y) if and only if we have the Cauchy–Riemann equations 1 2 = ∂ 1 ∂ 1 A Ξ− B Ξ− ∂ξ1 ◦ ∂ξ2 ◦ ∂ A Ξ 1 ∂ B Ξ 1 , " − # = " − # ∂ξ2 ¡ ◦ ¢ − ∂ξ1¡ ◦ ¢ ¡ ¢ ¡ ¢ which could be transformed to the desired system (2.8) via the chain rule.

B E F A Remark 2.7 (Beltrami equations, [4]). The system x ω ω y is the Bel- B = − F G A · y ¸ · ω ω ¸·− x¸ trami equations associated to the metric gΣ Ed x2 2Fdxdy Gd y2 with ω pEG F 2. = + + = −

3. GENERALIZED GAUSS MAP AND OSSERMAN SYSTEM OF THE FIRST ORDER The main point of this section is to build our Osserman system of the ﬁrst order, which generalizes the Cauchy–Riemann equations, and to provide the proof of Theorem 3.8, which illustrates the birth of the Osserman system. As in [8, 16, 32, 34], we introduce the generalized Gauss map of minimal surfaces in R4. Inside the complex projective space CP3, we prepare the complex hyperquadric

Q {[z : z : z : z ] CP3 z 2 z 2 z 2 z 2 0}. 2 = 1 2 3 4 ∈ | 1 + 2 + 3 + 4 = Definition 3.1 (Generalized Gauss map of minimal surfaces in R4, [34, Section 2]). Let Σ2 be a minimal surface in R4. Consider a conformal harmonic immersion X : Σ R4, → ξ X (ξ). The generalized Gauss map of Σ is the map G : Σ Q CP3 defined by 7→ → 2 ⊂ ∂X ∂X ∂X G (ξ) i Q2. = ∂ξ = ∂ξ + ∂ξ ∈ " # · 1 2 ¸ The conformality of the immersion X guarantees that the generalized Gauss map is a well-defined Q2-valued function. The harmonicity of the immersion X guarantees that the generalized Gauss map is anti-holomorphic.

Lemma 3.2 (Generalized Gauss map of two dimensional minimal graphs in R4). Let

Σ2 x, y, f (x, y), g(x, y) R4 (x, y) Ω = ∈ | ∈ ©¡ ¢ ª 8 HOJOO LEE denote a minimal graph in R4 withthe metricEdx2 2Fdxdy Gd y2 and ω pEG F 2. + + = − Its generalized Gauss map G : Ω Q CP3 can be explicitly given in terms of the coordi- → 2 ⊂ nates (x, y) : G (x, y) [z : z : z : z ] = 1 2 3 4 G F G F G F : i : fx i fy : gx i gy = ω − ω ω + − ω ω + − ω · µ ¶ µ ¶ ¸ F E F E F E 1 i : i : 1 i fx i fy : 1 i gx i gy . = − ω ω − ω + ω − ω + ω · µ ¶ µ ¶ ¸ Proof. For the details of the deduction of Lemma 3.2, we refer to [25, Proposition 6], which was inspired by the equality in [31, Lemma, p. 290].

Deﬁnition 3.3 (Degenerate minimal surfaces in R4, [34, Section 2, p. 122]). We say that 4 a minimal surface Σ in R is degenerate if the image of its Q2-valued generalized Gauss map lies in a hyperplane of the complex projective space CP3.

Remark 3.4 (Gauss maps and representation of degenerate minimal surfaces in R4). For a geometric illustration of generalized Gauss map of degenerate minimal surfaces, we refer to [7, Figure 1]. As known in [16, Theorem 4.7], degenerate minimal surfaces in R4 could be described by an explicit representation analogous to the Enneper-Weierstrass representation formula for minimal surfaces in R3.

Remark 3.5 (Degeneracy of entire two dimensional minimal graphs in arbitrary codimensions, [34, Chapter 5]). Extending Bernstein’s Theorem that the only entire minimal graphs in R3 are planes, Osserman established that the generalized Gauss map of entire n 2 4 two dimensional minimal graphs in R + ≥ are degenerate. He also determined an explicit representationformula for entire two dimensional minimal graphs in R4 in terms of a single non-vanishing holomorphic function. Though Osserman’s complete non-planar minimal surfaces in R4 are entire graphs, they are not stable, except the holomorphic curves. Micallef [28, Corollary 5.1] established that if an entire two dimensional non- planar minimal graph in R4 is stable, then it is a holomorphic curve in C2.

Deﬁnition 3.6 (Osserman system). Let Σ be the graph in R4 of the pair f (x, y), g(x, y) of height functions deﬁned on the domain Ω: ¡ ¢ Σ Φ(x, y) x, y, f (x, y), g(x, y) R4 (x, y) Ω . = = ∈ | ∈ We recall that the induced© metric¡gΣ and the area element¢ on the surfaceª Σ reads

2 2 gΣ Ed x 2Fdxdy Gd y , d AΣ ωdxdy, ω EG F 2, = + + = = − where the coefficients of the first fundamental form are determinedp by E Φ Φ 1 f 2 g 2, F Φ Φ f f g g , G Φ Φ 1 f 2 g 2. = x · x = + x + x = x · y = x y + x y = y · y = + y + y Given a constant µ R {0}, we introduce ∈ − f E F g (3.1) x µ ω ω y , f = F G g · y ¸ · ω ω ¸·− x ¸ MINIMAL SURFACE SYSTEM IN EUCLIDEAN FOUR-SPACE 9 or equivalently, 1 E F gx ω ω fy (3.2) F G , gy = −µ fx · ¸ · ω ω ¸·− ¸ which will be called the Osserman system with the coefficient µ R {0}. ∈ − Remark 3.7. The definition ω pEG F 2 yields the formula = − 1 E F − G F ω ω ω − ω . F G = F E · ω ω ¸ ·− ω ω ¸ One can use this to check that the two systems (3.1)and(3.2) are equivalent to each other. Theorem 3.8 (Minimality and degeneracy of Osserman graphs in R4). When the pair f (x, y), g(x, y) satisfies the Osserman system (3.1) with the coefficient µ R {0}, the ∈ − graph Σ x, y, f (x, y), g(x, y) R4 (x, y) Ω is minimal in R4. Moreover, its general- ¡ = ¢ ∈ | ∈ ized Gauss map lies on the hyperplane z iµz 0 of the complex projective space CP3. ©¡ ¢ 3 + 4ª= Proof. To show the minimality of the graph Σ x, y, f (x, y), g(x, y) R4 (x, y) Ω , we = ∈ | ∈ employ Theorem 2.1. Indeed, we use the equalities (3.2) to obtain ©¡ ¢ ª 1 ∂ G F ∂ F E Σ f fx fy fx fy △ = ω ∂x ω − ω + ∂y −ω + ω · µ ¶ µ ¶ ¸ 1 ∂ ∂ µgy µgx = ω ∂x + ∂y − · ¸ 0, ¡ ¢ ¡ ¢ = and use the equalities in (3.1) to obtain 1 ∂ G F ∂ F E Σ g gx gy gx gy △ = ω ∂x ω − ω + ∂y −ω + ω · µ ¶ µ ¶ ¸ 1 ∂ 1 ∂ 1 fy fx = ω ∂x −µ + ∂y µ · µ ¶ µ ¶ ¸ 0. = To prove the degeneracy of the minimal graph Σ, we exploit Lemma 3.2. Its generalized Gauss map G : Ω Q CP3 can be explicitly given in terms of the coordinates (x, y) : → 2 ⊂ G F G F G F G (x, y) [z1 : z2 : z3 : z4] : i : fx i fy : gx i gy , = = ω − ω ω + − ω ω + − ω · µ ¶ µ ¶ ¸ The Osserman systems (3.1)and(3.2) yield F G 1 G F fy , gy µ gy gx , fx fy , = ω − ω µ ω − ω µ µ ¶ µ ¶¶ which can be complexified¡ to¢ G F G F (3.3) fx i fy iµ gx i gy . ω + − ω = − ω + − ω µ ¶ µ µ ¶ ¶ Therefore, by the above formula, its generalized Gauss map G lies on the hyperplane z iµz 0. 3 + 4 = 10 HOJOO LEE

4. APPLICATIONS OF LAGRANGEPOTENTIALONMINIMALGRAPHSIN R3 It would be difficult to find explicit examples of non-holomorphic minimal graphs in R4 by directly solving the minimal surface system of the second order. We provide a fundamental construction, which yields explicit examples of two dimensional minimal graphs in R4, whose height functions satisfy the Osserman system of the first order. Lemma 4.1 (Lagrange potential on minimal graphs in R3). Let Ω R2 be a simply con- ⊂ nected domain. Consider the two dimensional graph S of the C 2 function p : Ω R : → S x, y, p(x, y) R3 (x, y) Ω . = ∈ | ∈ Then, the following two statements©¡ are equivalent:¢ ª (a) The graph S is a minimal surface in R3. (b) There exists a function q : Ω R satisfying the Lagrange system →

py px (4.1) qx, qy  , , = − 2 2 2 2 1 px py 1 px py ¡ ¢  + + + +  and the gradient estimate q q  (4.2) q 2 q 2 1. x + y < Proof. The graph S is minimal in R3 if and only if the height function p(x, y) satisﬁes

p ∂ px ∂ y Ω (4.3) 0    , x, y , = ∂x 2 2 + ∂y 2 2 ∈ 1 px py 1 px py  + +   + +  ¡ ¢ which indicates that theq one form  q  py px (4.4) d x d y − 1 p 2 p 2 + 1 p 2 p 2 + x + y + x + y is closed. Since Ω is simplyq connected, by Poincaréq Lemma, it means that (4.4) is exact: py px d x d y dq qxd x qy d y − 1 p 2 p 2 + 1 p 2 p 2 = = + + x + y + x + y for some potentialq function q : Ω qR, up to an additive constant. The gradient estimate → 2 2 1 (4.2) immediately follows from the identity 1 qx qy 1 p 2 p 2 . − − = + x + y Remark 4.2 (Lagrange potentials and conjugate surfaces of minimal graphs in R3). The exactness of the one form (4.4) on the minimal graph is discovered by Lagrange [22], who deduced the minimal surface equation (4.3). The Lagrange potential plays a critical role in the existence theory of Jenkins-Serrin minimal graphs [17, Section 3] obtained by solving the minimal surface equation with infinite boundary value problems. In Theorem 4.3, we shall use the Lagrange potential to build a deformation of minimal graphs in R3 defined on a domain to degenerate minimal graphs in R4 defined on the same domain. As a particular case of the system (2.8) in Proposition 2.6, when the graph S x (x, y), x (x, y), x (x, y) x, y, p(x, y) R3 (x, y) Ω = 1 2 3 = ∈ | ∈ ©¡ ¢ ¡ ¢ ª MINIMAL SURFACE SYSTEM IN EUCLIDEAN FOUR-SPACE 11

3 is minimal in R , we ﬁnd that, for each k {1,2,3}, the function x i x∗ is holomorphic ∈ k + k on S, whenever we have the Cauchy-Riemann equations on the minimal graph S:

2 px py 1 p + x x 2 2 2 2 k∗ x p1 px py −p1 px py (xk)x (4.5) + +2 + + . x  1 py px py  k∗ y = + (xk)y "¡ ¢ # 2 2 2 2 · ¸ p1 px py −p1 px py  + + + +  ¡ ¢   3 (a) The conjugate surface S∗ x∗(x, y), x∗(x, y), x∗(x, y) R (x, y) Ω becomes = 1 2 3 ∈ | ∈ a minimal surface locally isometric to S. According to Krust Theorem [9, p. 122], ©¡ ¢ ª whenever the domain Ω of the initial minimal graph S is convex, its conjugate minimal surface S∗ can be represented as a graph on some domain Ω∗. (b) Taking k 3 in (4.5) yields the Lagrange system (4.1), which is equivalent to = 2 px py 1 p + x 2 2 2 2 qx p1 px py −p1 px py px (4.6) + +2 + + ,  1 py px py  qy = + py · ¸ 2 2 2 2 · ¸ p1 px py −p1 px py  + + + +    The function p iq is holomorphic on S (with respect to the classical conformal + coordinates constructed in Proposition 2.6). (c) We combine the gradient estimation (4.2) and the Lagrange system (4.1) to deduce

∂ qx ∂ qy ∂ ∂ (4.7)     py px 0, ∂x 2 2 + ∂y 2 2 = ∂x − + ∂y = 1 qx qy 1 qx qy  − −   − −  ¡ ¢ ¡ ¢ q  q  or equivalently,

(4.8) 1 q 2 q 2q q q 1 q 2 q 0. − y xx + x y xy + − x yy = As a historical remark,¡ ¢ the dual equation¡ (4.8) is¢ reported in 1855 by Catalan [6, Equation (C), p. 1020], wherehediscoveredhis minimal surfacegeneratedbyaone parameter family of parabolas and contains a cycloid as a geodesic. Calabi found that, in his article [5] on Bernsteintype problems, (4.2) andthedual equation(4.7) indicates that the graph z q(x, y) is a maximal surface (spacelike surface with = zero mean curvature)in Lorentz-Minkowskispace L3 R3,d x2 d y2 d z2 . The = + − author [25] extended the Calabi duality between minimal graphs in R3 and max- ¡ ¢ imal graphs in L3 to higher codimensions. (d) Taking k 1 and k 2 in (4.5) yields two identities = = 2 ∂ px py ∂ 1 py (4.9)    + , ∂y 2 2 = ∂x 2 2 1 px py 1 px py  + +   + +  q  q  and

2 ∂ 1 px ∂ px py (4.10)  +   . ∂y 2 2 = ∂x 2 2 1 px py 1 px py  + +   + +  q  q  12 HOJOO LEE

Following previous notations, these two equalities can be rewritten as ∂ F ∂ G ∂ E ∂ F (4.11) and . ∂y ω = ∂x ω ∂y ω = ∂x ω µ ¶ µ ¶ µ ¶ µ ¶

Theorem 4.3 (Degenerate minimal graphs in R4 derived from minimal graphs in R3). Let Σ be the minimal graph of the C 2 function p : Ω R deﬁned on a domain Ω R2 : 0 → ⊂ Σ x, y, p(x, y) R3 (x, y) Ω . 0 = ∈ | ∈ Let q : Ω R be the Lagrange potential©¡ in Lemma¢ 4.1, whichª solves the Lagrange system →

py px (4.12) qx, qy  , . = − 2 2 2 2 1 px py 1 px py ¡ ¢  + + + +  q q For any constant λ R {0}, we associate the two dimensional graph in R4: ∈ − (4.13) Σ x, y,(coshλ) p(x, y),(sinhλ) q(x, y) R4 (x, y) Ω . λ = ∈ | ∈ Then, the pair f (x,©¡y), g(x, y) (coshλ) p(x, y),(sinhλ)¢q(x, y) solves theª Osserman sys- = tem (3.1) with the coefficient µ cothλ. In particular,the graph Σ is minimal in R4. Also, ¡ ¢= ¡ ¢λ we obtain the conformal invariance of the conformally changed induced metric 1 1 (4.14) gΣ gΣ . λ = 0 det gΣλ det gΣ0 Proof. Our goal is to verify theq Osserman¡ ¢ systemq (3.1¡ ) with¢ the coefficient µ cothλ: = f E F g (4.15) x cothλ ω ω y . f = F G g · y ¸ · ω ω ¸·− x ¸ Taking W 1 p 2 p 2 1 and using the system (4.12), we have = + x + y ≥ q 2 py px 2 2 W 1 (4.16) qx, qy , and qx qy − . = − W W + = W 2 ³ ´ We use the definition¡ f , g)¢ (coshλ) p,(sinhλ) q to deduce = ω2 ¡ EG¢ ¡F 2 ¢ = − 1 f 2 g 2 1 f 2 g 2 f f g g 2 = + x + x + y + y − x y + x y 2 ¡1 f 2 f 2 ¢¡ g 2 g 2 ¢ f¡ g f g ¢ = + x + y + x + y + x y − y x 2 ¡ ¢ sinh¡ 2 λ ¢ ¡ ¢ cosh2 λ W . = − W · ¸ We observe that ¡ ¢ sinh2 λ sinh2 λ cosh2 λ W cosh2 λ 1 1 0, − W ≥ · − 1 = > which implies that¡ ¢ ¡ ¢ sinh2 λ (4.17) ω EG F 2 cosh2 λ W . = − = − W p ¡ ¢ MINIMAL SURFACE SYSTEM IN EUCLIDEAN FOUR-SPACE 13

We use (4.16) to obtain the ﬁrst row equality in (4.15):

E F 1 2 2 gy gx 1 fx gx gy fx fy gx gy gx ω − ω = ω · + + − + 1 £¡ 2 ¢ ¡ ¢ ¤ 1 fx gy fx fy gx = ω · + − sinh£¡λ px ¢ 2 ¤ 2 2 1 cosh λ px py = ω · W · + + sinhλ px £ ¡ ¢¡ ¢¤ sinh2 λ cosh2 λ W 2 = ω · W · − + (sinhλ) p £ ¡ ¢ ¤ = x f x . = cothλ We omit a similar veriﬁcation of the second row equality in (4.15). Finally, the conformal invariance (4.14) comes from the equalities 2 2 E F G 1 px px py 1 py , , + , , + . ω ω ω = W W W µ ¶ Ã !

We apply Theorem 4.3 to classical minimal graphs in R3 to ﬁnd explicit examples of old and new minimal graphs in R4. Example 4.4 (One parameter family of minimal surfaces in R4 foliated by hyperbolas Ω R2 R π π or lines). On the inﬁnite strip : x, y x and y 2 , 2 , we consider the = R3 ∈ | ∈ ∈ − fundamental piece of the helicoid in©¡ : ¢ ¡ ¢ª Σ x, y, x tan y R3 (x, y) Ω . 0 = ∈ | ∈ Solving the induced Lagrange system©¡ (4.1) in¢ Lemma 4.1 ª

px py cos y sin y x qy , qx  ,  , , − = 2 2 2 2 = 2 2 2 2 1 px py 1 px py Ã cos y x cos y x ! ¡ ¢  + + + +  + + q q  p p we obtain p(x, y) x tan yq(x, y) cos2 y x2 , up to an additive constant. Applying = = − + Theorem 4.3 with any constant λ R, we obtain the solution of the Osserman system: ∈ p f (x, y), g(x, y) (coshλ) x tan y, sinhλ cos2 y x2 , = − + µ q ¶ ¡ ¢ Σ R4 which associates, after an reﬂection, the two dimensional minimal graph λ− in :

2 2 4 Σ− x, y,(coshλ) x tan y, sinhλ cos y x R (x, y) Ω . λ = + ∈ | ∈ ½µ q ¶ ¾ 3 (a) When λ 0, the graph Σ− recovers the helicoid in R foliated by lines. = λ (b) Let λ 0. We see that the graph Σ− is foliated by a line (the level curve cut by the 6= λ hyperplane y 0) or a hyperbola (the level curve cut by the hyperplane y y = = 0 ∈ π ,0 0, π ). − 2 ∪ 2 ¡ ¢ ¡ ¢ 14 HOJOO LEE

Under the coordinate transformation (x, y) (sinhU cosV ,V ) (U ,V ), we obtain the = Σ R4 → conformal harmonic patch for the minimal surface λ− in : (4.18) F− (U ,V ) (sinhU cosV ,V ,coshλcoshU cosV ,sinhλsinhU sinV ). θ = Σ The graph λ− belongs to the family of minimal surfaces discovered by the author [26, Example 6.1]. It was originally discovered by an application of the so called parabolic rotations of holomorphic null curves in C3 C4 lifted from helicoids in R3. ⊂ Example 4.5 (Hoffman-Osserman’s minimal surfaces in R4 foliated by ellipses). We consider a half of the catenoid in R3:

Σ x, y, x2 cosh2 y R3 (x, y) Ω , 0 = − + ∈ | ∈ ½µ q ¶ ¾ over the domain (4.19) Ω : x, y R2 x2 cosh2 y , = ∈ | ≤ which can be compared with the©¡exceptional¢ domain [38,ª Section 7.2 and Figure 1]. We ﬁnd the pair p(x, y), q(x, y) x2 cosh2 y , x tanh y solving the Lagrange system = − + µ ¶ ¡ ¢ q p py x sinh y x − qy , qx  ,  2 , . − = 2 2 2 2 = cosh y cosh y 1 px py 1 px py µ ¶ ¡ ¢  + + + +  q q  4 Applying Theorem 4.3 with a constant λ R {0}, we obtain the minimal graph Σ+ in R : ∈ − λ 2 2 4 Σ+ x, y,coshλ x cosh y ,(sinhλ) x tanh y R (x, y) Ω . λ = − + ∈ | ∈ ½µ q ¶ ¾ Under the coordinate transformation (x, y) (coshU cosV ,U ) (U ,V ), we obtain the = Σ R4 → conformal harmonic patch for the minimal surface λ+ in :

(4.20) F+ (U ,V ) (coshU cosV ,U ,coshλcoshU sinV ,sinhλsinhU cosV ). λ = This recovers Osserman-Hoffman’s minimal annuli in R4 with total curvature 4π ([16, − Proposition 6.6 and Remark 1] and [26, Example 6.2 and Theorem 6.3]). Remark 4.6 (Holomorphic null curves lifted from degenerate minimal graphs in R4). 3 In Theorem 4.3, if the initial minimal graph Σ0 in R is induced by the holomorphic null curve φ φ (ζ),φ (ζ),φ (ζ) in C3 with φ 2 φ 2 φ 2 0 and a local conformal coor- = 1 3 3 1 + 2 + 3 = dinate ζ on Σ , the minimal graph Σ in R4 is induced by ¡ 0 ¢ λ φ (ζ),φ (ζ),(coshλ)φ (ζ),( i sinhλ)φ (ζ) 1 2 3 − 3 with the conformal coordinate¡ ζ on Σλ. Indeed, we see that the identity¢ (coshλ)2 ( i sinhλ)2 1 + − = guarantees the nullity of the induced holomorphic curve in C4 2 2 φ 2 φ 2 (coshλ)φ ( i sinhλ)φ φ 2 φ 2 φ 2 0. 1 + 2 + 3 + − 3 = 1 + 2 + 3 = For a concise survey of various£ deformations¤ £ of holomorphic¤ null curves in Cn lifted from minimal surfaces in Rn, we invite interested readers to refer to [26, Section 2]. MINIMAL SURFACE SYSTEM IN EUCLIDEAN FOUR-SPACE 15

Remark 4.7 (Associate family of minimal surfaces in higher codimensions). We point out that the conformal harmonic patches (4.20) in Example 4.5 and (4.18) in Example 4.4 represent conjugate minimal surfaces in R4. For the notion of associate family of locally n 2 3 isometric minimal surfaces in R + ≥ , we invite interested readers to refer to [23]. Example 4.8 (Doubly periodic minimal graphs in R4 derived from Scherk’s doubly pe- R3 Ω R2 π π riodic graph in ). Over the open square : x, y x, y 2 , 2 , we define the = R3 ∈ | ∈ − fundamental piece of the doubly periodic graph©¡ in ¢: ¡ ¢ª cos x (4.21) x, y,ln R3 (x, y) Ω , cos y ∈ | ∈ ½µ µ ¶¶ ¾ which was originally discovered by Scherk [37, p. 196]. See [27] for the picture of the surface. Theorem 4.3 with a constant λ R {0} gives the minimal graph in R4: ∈ − cos x x, y,(coshλ)ln ,(sinhλ)arcsin sin x sin y R4 (x, y) Ω . cos y ∈ | ∈ ½µ µ ¶ ¶ ¾ ¡ ¢ Remark 4.9 (Finn-Osserman proof of Bernstein’s Theorem, [12]). We point out that Finn and Osserman used Scherk’s first surface to present an elegant geometric proof of Bernstein’s Theorem that they only entire minimal graphs in R3 are planes. Remark 4.10 (Jenkins-Serrin type minimal graphs). Inspired by the existence of Scherk’s first surfaces, Jenkins and Serrin [17] offers a powerful analytic method to extend Scherk’s construction. The fundamental piece of Scherk’s first surface can be obtained as a Jenkins- Serrin graph by solving the Dirichlet problem for the minimal surface equation over a square and taking boundary values plus infinity on two opposite sides and minus infinity on the other two opposite sides. See also Karcher’s beautiful paper [20, Section 2.6.1]. Remark 4.11 (Lagrangian Scherk graph in R4, [25, Example 3]). It would be interesting to develop the Jenkins-Serrin type construction for minimal surface system. We consider Σ x, y,arsinh tanx cos y ,arsinh tan y cos x R4 (x, y) Ω , = ∈ | ∈ where Ω is the domain©¡ of the¡ Scherk’s graph¢ (4.21¡) in Example¢¢ 4.8. Due to theª equality ∂ ∂ arsinh tanx cos y arsinh tan y cos x , ∂y = ∂x we find that the minimal¡ surface¡ Σ is a gradient¢¢ graph.¡ Taking¡ the function¢¢ h(x, y) with h ,h arsinh tanx cos y ,arsinh tan y cos x , x y = one deduce the Monge-Ampére¡ ¢ ¡ equation¡ ¢ ¡ ¢¢ h h h 2 1, xx yy − xy = a particular case of special Lagrangian equation with the phase ([14] and [40, Section 1]). Since Σ is special Lagrangian in C2, (as in [19, Section 8.1.1]), it is a holomorphic curve with respect to other orthogonal complex structure on R4. Taking the complexification (ζ ,ζ ) x ih , y ih , 1 2 = + y − x we find that, h h h 2 1 implies that¡ the complex¢ curve Σ is holomorphic: xx yy − xy = dζ dζ i h h h 2 1 d x d y 0. 1 ∧ 2 = − xx yy − xy − ∧ = ¡ ¢ 16 HOJOO LEE

Example 4.12 (Doubly periodic minimal graphs in R4 derived from sheared Scherk’s doubly periodic graph in R3). More generally, Scherk [37, p. 187] showed that his surface (4.21) in Example 4.8 belongs to a one parameter family of doubly periodic minimal graphs. Following [30, p. 70], for an angle constant 2α (0,π) and a dilation constant ∈ ρ 0, we define Scherk’s doubly periodic minimal graph Σ2α by > ρ ρ y 1 cos x 2 cosα − sinα (4.22) z ln ρ x y , = ρ "cos¡ £ ¤¢# 2 cosα + sinα Ω where its domain is an infinite chess board-like¡ £ net of rhombo¤¢ ids i,j ZRij . Here, R π = ∪ ∈ we define the rhomboid domain ij with the length ρ as follows

2 x y 4i π x y 4j π Rij (x, y) R : π , π . = ∈ cosα − sinα − ρ < ρ cosα + sinα − ρ < ρ ½ ¯ ¯ ¯ ¯ ¾ ¯ π ¯ ¯ ¯ π ¯ Σ 2 ¯ ¯ ¯ Taking α 4 and ρ 2 in¯ (4.22), the graph 2 ¯is congruent¯ to the minimal¯ graph (4.21) = = π R4 in Example 4.8, after the 4 -rotation. Theorem 4.3 gives the minimal graph in : x, y,(coshλ) p(x, y),(sinhλ) q(x, y) R4 (x, y) Ω , ∈ | ∈ where the pair (p©¡(x, y), q(x, y)) of functions is determined¢ by ª

ρ x y 1 cos 2 cosα sinα p(x, y) ln ρ − y , ρ cos x = ¡ 2 £ cosα + sinα ¤¢  · ¸ρx ρy q(x, y) 1 arccos¡ cos£ 2 αcos¤¢ sin2 αcos .  = ρ cosα − sinα Example 4.13 (Minimal graphs in£R4 derived¡ from¢ Scherk’s saddle¡ ¢¤ tower in R3). Over the domain Ω : x, y R2 1 sinh x sinh y 1 , we consider a fundamental piece = ∈ | − < < of the singly periodic multi-valued graph in R3: ©¡ ¢ ª (4.23) x, y,arcsin sinh x sinh y R3 (x, y) Ω , ∈ | ∈ which was originally discovered©¡ by¡ Scherk [37,¢¢ p. 198]. See [27ª] for the picture of this surface. Theorem 4.3 with a constant λ R {0} gives the minimal graph in R4: ∈ − cosh x x, y,(coshλ)arcsin sinh x sinh y ,(sinhλ)ln R4 (x, y) Ω . cosh y ∈ | ∈ ½µ µ ¶¶ ¾ ¡ ¢ Remark 4.14 (Scherk’s saddle tower in R3 and its influences). Geometrically, Scherk’s saddle tower is a smooth minimal desingularizationof two perpendicular vertical planes. Though it was discovered more than 180 years ago, it still plays a fundamental role in the modern theory of desingularizations and gluing construction for surfaces with constant mean curvature and solitons to various curvature flows. Inspired by Scherk’s saddle tower, Karcher [20] discovered embedded minimal surfaces in R3 derived from Scherk’s examples, with explicit Weierstrass datum, and Pacard [35] showed the existence of (N 2)-periodic N 4 − embedded minimal hypersurfaces in R ≥ with four hyperplanar ends. Example 4.15 (Minimal graphs in R4 derived from Scherk’s generalized tower in R3). We take the fundamental piece of the singly periodic multi-valued graph in R3: 1 ρx ρy (4.24) z p(x, y) arccos cos2 αcosh sin2 αcosh = = ρ cos α − sinα h ³ ´ ³ í MINIMAL SURFACE SYSTEM IN EUCLIDEAN FOUR-SPACE 17 following [35, Section 1] and [30, p. 74]. Theorem 4.3 produces the minimal graph in R4 x, y,(coshλ) p(x, y),(sinhλ) q(x, y) R4 (x, y) Ω , ∈ | ∈ where we take the©¡ function ¢ ª ρ y 1 cosh x 2 cosα − sinα q(x, y) ln ρ x y . = ρ "cosh¡ £ ¤¢ # 2 cosα + sinα ¡ £ ¤¢ 5. MINIMAL GRAPHS IN R3 AND SPECIAL LAGRANGIAN GRAPHS IN R6 C3 = Inspired by Bernstein’s Theorem that the entire minimal graphs in R3 are planar, Fu [14], Jost-Xin [18], Tsui-Wang [39], Yuan [41] established Bernstein type results for entire special Lagrangian graphs in even dimensional Euclidean space. Under suitable analytic or geometric assumptions, entire special Lagrangian graphs should be planar.

Theorem 5.1 (Special Lagrangian equation in C3, [15, Theorem 2. 3 and (4.8)]). Let S be the gradient graph of the R-valued function F(x, y, z) in R3. Then, the 3-fold S in R6 admits an orientation making it into a special Lagrangian graph in C3 under the complexiﬁcation x i y , x i y , x i y x , x , x , y , y , y 1 + 1 2 + 2 3 + 3 = 1 2 3 1 2 3 when the function F¡(x, y, z) satisﬁes the special¢ Lagrangian¡ equation¢

Fxx Fxy Fxz Fxx Fxy Fxz (5.1) det Fyx Fyy Fyz tr Fyx Fyy Fyz .   =   Fzx Fzy Fzz . Fzx Fzy Fzz     Remark 5.2. In [15, III.4.B. Degenerate projections and harmonic gradients], Harveyand Lawson investigated the interesting special case when the solution of the equation (5.1) is afﬁne with respect to the coordinate z. The function F(x, y, z) p(x, y) z h(x, y) satisﬁes = + the special Lagrangian equation (5.1) if and only if the pair p,q solves the system

2 2 ¡ ¢ 1 py pxx 2px py pxy 1 px pyy 0, + 2 − + + 2 = ( 1 py hxx 2px py hxy 1 px hyy 0. ¡ + ¢ − +¡ + ¢ = These two partial differential¡ equations¢ admit interestin¡ g¢ geometric interpretations. The ﬁrst equation means that the graph of p(x, y) is a minimal surface in R3, and the second equation means that q(x, y) is harmonic on the graph of p(x, y). In this case, the patch of the gradient graph of F(x, y, z) p(x, y) z h(x, y) is also afﬁne with respect to z: = + x x x 0 y y y 0 x         Φ z z 0 1 y       z  ,   = Fx = px (x, y) zqx (x, y) = px (x, y) + qx (x, y) z    +      Fy  py (x, y) zqy (x, y) py (x, y) qy (x, y)      +      Fz   q(x, y)   q(x, y)   0                  which geometricallymeans that the gradient graph is generatedbylines z R Φ x, y, z . ∈ 7→ ¡ ¢ 18 HOJOO LEE

As the second application of the Lagrange potential on minimal graphs in R3, we use the above Harvey-Lawson construction to build a one parameter family of special La- grangian graphs in C3 derived from minimal graphs in R3. Theorem 5.3 (Lagrange potential construction of special Lagrangian graphs in R6 C3). = We consider the minimal graph of the function p(x, y) : Ω R on the domain Ω R2 : → ⊂ Σ x, y,p(x, y) R3 (x, y) Ω . = ∈ | ∈ Let q(x, y) : Ω R be the Lagrange©¡ potential,¢ which satisﬁesª the Lagrange system →

py px qx ,qy  , . = − 2 2 2 2 1 px py 1 px py ¡ ¢  + + + +   q q  For any constant λ R, we associate the three dimensional graph Σ in R6 ∈ λ p y px R6 Ω R x, y, z, px λz , py λz , λq (x, y) , z . − 2 2 + 2 2 ∈ | ∈ ∈  1 px py 1 px py   + + + +   Σ q q C3  Then, the 3-fold λ becomes a special Lagrangian graph in .  Proof. By the item (c) in Remark 4.2, the function p iq is holomorphic on the minimal + graph Σ. Since p and q are harmonic on the minimal graph Σ, by Remark 2.2, we obtain 2 2 1 py pxx 2px py pxy 1 px pyy 0, (5.2) + − + + = 1 p 2 λq 2p p λq 1 p 2 λq 0. (¡ + y ¢ xx − x y ¡ xy + ¢+ x yy = However, by [15, Theorem¡ ¢¡ 4.9],¢ the system¡ (5.2¢) guarantees¡ ¢¡ that¢ the gradient graph of the function F(x, y, z) p(x, y) λz q(x, y) becomes a special Lagrangian 3-fold in C3. = + Remark 5.4. The construction in Theorem 5.3 with λ 0 recovers the product of the line = R and the special Lagrangian 2-fold given by the gradient graph of the function p(x, y). Example 5.5 (Doubly periodic special Lagrangian graph in C3). We apply Theorem 5.3 to the fundamental piece of Scherk’s doubly periodic graph on the domain Ω in Example 3 4.8 to have the one parameter family of special Lagrangian graph Σλ in C : Σ x, y, z, A x, y, z ,B x, y, z ,C x, y, z R6 (x, y) Ω, z R , λ = ∈ | ∈ ∈ ©¡ ¡ sin¢ x cos¡ y ¢sin y¡ ¢¢ cosx sin y ª where (A,B,C) sinx λz , λz , λarcsin sin x sin y . = − cosx + p1 sin2 x sin2 y cos y + p1 sin2x sin2 y µ − − ¶ ¡ ¢

Acknowledgements. The author met Finn-Osserman’s elegant proof [12] of Bern- stein’s Theorem in R3, which uses Scherk’s doubly periodic minimal graph (Remark 4.9), during a lecture presented by Antonio Ros at the international conference held in Seville, Spain. Part of this work was carried out while he was visiting Jaigyoung Choe at Korea In- stitute for Advanced Study, Seoul, Korea. He would like to thank KIAS for the hospitality. He would like to warmly thank J. C. (again) who presented him with a copy of “A Survey of Minimal Surfaces”when he was an ǫ. MINIMAL SURFACE SYSTEM IN EUCLIDEAN FOUR-SPACE 19

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