Reflection principle for lightlike line segments on maximal surfaces
aa r X i v : . [ m a t h . DG ] F e b REFLECTION PRINCIPLE FOR LIGHTLIKE LINE SEGMENTSON MAXIMAL SURFACES
SHINTARO AKAMINE AND HIROKI FUJINO
Abstract.
As in the case of minimal surfaces in the Euclidean 3-space, thereflection principle for maximal surfaces in the Lorentz-Minkowski 3-space assertsthat if a maximal surface has a spacelike line segment L , the surface is invariantunder the 180 ◦ -rotation with respect to L . However, such a reflection propertydoes not hold for lightlike line segments on the boundaries of maximal surfaces ingeneral.In this paper, we show some kind of reflection principle for lightlike line seg-ments on the boundaries of maximal surfaces when lightlike line segments areconnecting shrinking singularities. As an application, we construct various exam-ples of periodic maximal surfaces with lightlike lines from tessellations of R . Introduction
The classical Schwarz reflection principle for harmonic functions yields a sym-metry principle for minimal surfaces in the 3-dimensional Euclidean space E : ifa minimal surface in E has a straight line segment L on its boundary, then thesurface can be extended across L and the extended surface is invariant under the180 ◦ -rotation with respect to L (see [6, p. 289], [24, p. 140] and [25, p. 54] forexample). This principle is directly derived from the Schwarz reflection principleand the fact that each coordinate function of a conformal minimal immersion in E is harmonic. For the same reason, such a reflection principle for lines also holdsfor maximal surfaces, i.e. spacelike surfaces with vanishing mean curvature, in the3-dimensional Lorentz-Minkowski space L , when the straight line segment is space-like, see Al´ıas-Chaves-Mira [5, Theorem 3.10]. As a singular version of this reflectionprinciple, a reflection principle inducing point symmetries for shrinking or conelikesingularities was also shown in [10, 11, 21, 23] (see also [12, 15, 17, 22]).These regular and singular versions of reflection principles for maximal surfacesare highly depend on the conformal structures of surfaces. However, as anotherpossibility of a line reflection principle, lightlike line segments can appear on theboundaries of maximal surfaces. Unfortunately, it was shown in the author’s pre-vious work [3] that these boundary lightlike line segments appear as discontinuous Date : February 20, 2020.2010
Mathematics Subject Classification.
Primary 53A10; Secondary 53B30; 31A05; 31A20.
Key words and phrases. reflection principle, maximal surface, lightlike boundary problem, har-monic mapping.The first author was partially supported by JSPS KAKENHI Grant Number 19K14527,17H06466 and JSPS/FWF Bilateral Joint Project I3809-N32 “Geometric Shape Generation”, andthe second author by JSPS KAKENHI Grant Number 19K21022. boundary behaviors of conformal maximal immersions and hence conformal struc-tures break down on these lines. Therefore, we cannot expect a conventional sym-metry principle for lightlike line segments on maximal surfaces in general. In fact,a maximal surface with lightlike line segments along which neither a line symmetrynor planar symmetry holds was given in [3, Section 4.5]. In particular, as far as theauthors know, the following problem was still open.
Question 1.
Is there a reflection principle for boundary lightlike line segments ofmaximal surfaces?
We can also see a similar question in [20, p.1095] for boundary lightlike line segmentsfor timelike minimal surfaces in L .Answering Question 1, in the present article we show a reflection principle for aboundary lightlike line segment connecting two shrinking singularities: Theorem A.
If a maximal graph Σ has a lightlike line segment L ⊂ ∂ Σ connectingtwo shrinking singularities as the endpoints of L . Then Σ can be extended to a surfacewith zero mean curvature across L which is invariant under the point symmetry atthe midpoint of L . For a more precise statement, see Theorem 2.3. The situation in Theorem A isvery typical since a large number of maximal surfaces with lightlike line segmentshave shrinking singularities as their endpoints. See Figure 1 and Example 2.7.
Figure 1.
Examples of periodic maximal surfaces with lightlike lines(white lines) connecting shrinking singularities.Moreover, as an important application of Theorem A, we give a construction ofnew proper periodic maximal surfaces with lightlike line segments as in Figure 2.
Figure 2.
Doubly (center) and triply periodic (right) maximal sur-faces generated by the regular hexagonal tiling of R . EFLECTION PRINCIPLE FOR LIGHTLIKE LINE SEGMENTS 3
The organization of this article is as follows: In Section 2.1, we prepare a neces-sary condition for existence of extensions of maximal surfaces across lightlike linesegments. In Section 2.2, we explain how lightlike lines and shrinking singularitiesappear on the boundaries of maximal surfaces from aspects of harmonic functiontheory. In Section 2.3, we give a proof of Theorem A. This is achieved by func-tion theoretical “blow-up” method of discontinuous boundary points of harmonicmappings, instead of the Schwarz reflection principle, which is a theorem aboutcontinuous boundary points of harmonic functions. In Section 3, we give a construc-tion procedure of some periodic maximal surfaces with lightlike line segments byusing Theorem A and tessellations of R associated with symmetries appearing inTheorem A. Finally, we discuss in Section 4 the remaining cases, that is, reflectionprinciples for entire lightlike lines and lightlike half-lines as a future question.2. blow-up of discontinuous boundary points In this paper, L denotes the Lorentz-Minkowski 3-space with signature (+ , + , − ).Throughout this section, unless otherwise noted, we let Σ = graph( ψ ) ⊂ L be amaximal graph, defined by a smooth function ψ : Ω → R over a simply connectedbounded Jordan domain Ω in the xy -plane. Here, we assume that Σ contains alightlike line segment in its boundary. In this section, we discuss an extension of Σacross the boundary lightlike line segment.2.1. A necessary condition.
For the existence of such an extension, a necessarycondition was introduced in [3, Definition 3.1]: Let I ⊂ ∂ Ω be an open Euclideansegment, and τ ∈ R the unit tangent vector to I with the positive direction. Here,the positive direction is determined by the counterclockwise orientation of ∂ Ω. Wesay that ψ tamely degenerates to a future (resp. past)-directed lightlike line segmentover I if ψ satisfies ∂ψ∂τ ( x, y ) = 1 + O (dist (cid:0) ( x, y ) , J ) (cid:1) (cid:18) resp . ∂ψ∂τ ( x, y ) = − O (dist (cid:0) ( x, y ) , J ) (cid:1)(cid:19) as ( x, y ) → J for each closed segment J ⊂ I , where ∂/∂τ denotes the directionalderivative in the τ -direction. We simply say that ψ tamely degenerates to a lightlikeline segment if it tamely degenerates to a future or past-directed lightlike line seg-ment. In this case, it is easily seen that ∂ Σ contains a lightlike line segment over I .Conversely, if ψ extends to a C -function across I and its graph contains a lightlikeline segment L over I , then ψ tamely degenerates to L (see [3, Remark 3.3]). There-fore, ψ needs to tamely degenerate to a lightlike line segment if Σ extends acrossthe lightlike line segment.2.2. Lightlike line with shrinking singularities.
Let X : D → Σ = graph( ψ )be an isothermal parametrization from the unit disk D = { w ∈ C | | w | < } . Thespacelike condition of Σ implies that Σ is bounded since ψ is a locally 1-Lipschitzfunction over a bounded domain. Thus, X is a bounded harmonic mapping, andtherefore can be written as a Poisson integral of some bounded mapping b X : ∂ D → L (see [19, p.72, Lemma 1.2]), that is, X ( w ) = P D b X ( w ) := 12 π Z π − | w | | e it − w | b X ( e it ) dt. S. AKAMINE AND H. FUJINO
It is well-known that if w ∈ ∂ D is a jump point of b X , that is, the one-sided limits a := ess . lim Let L ⊂ ∂ Σ be a lightlike line segment and a, b ∈ L ( a = b ) itsendpoints. We say that L has shrinking singularities at the endpoints if there exists w = e it ∈ ∂ D such that b X ( e it ) ≡ a for all t ∈ ( t − ε, t ) and b X ( e it ) ≡ b for all t ∈ ( t , t + δ ) for some ε, δ > Remark . The map b X is determined up to sets of measure zero. Thus, Definition2.1 requires that b X satisfies the conditions after an appropriate modification on aset of measure zero.In this case, the isothermal parametrization X extends to a harmonic mapping acrosseach of the two arcs ( t − ε, t ) and ( t , t + δ ) by the Schwarz reflection principle.The extended map becomes a generalized maximal surface in the sense of [8] and hasshrinking singularities at a and b . Here, the definitions of the generalized maximalsurfaces and the shrinking singularities were introduced in [8] and [21], respectively.Furthermore, by [3, Corollary 3.7], we can see ψ tamely degenerates to L .There are useful ways to check whether the endpoints of a lightlike line segmentare shrinking singularities or not from the shape of Σ. Such methods will be givenin Section 2.4.2.3. Reflection principle for lightlike line segments. Recall that a lightlikeline segment L ⊂ ∂ Σ corresponds to a single point under an isothermal parametriza-tion. Thus, this isothermal parametrization seems to be quite useless to constructan extension of Σ across L . However, we introduce a modification method of theisothermal parameter like a “blow-up”, and consequently we will prove that Σ canbe extended across L with some reflection symmetry if L has shrinking singularitiesat its endpoints.To explain the modification, it is useful to consider the Poisson integral for theupper half plane H = { ζ ∈ C | Im( ζ ) > } . For a bounded piecewise continuousfunction U : R → R , the Poisson integral is defined by P U ( ζ ) = 1 π Z ∞−∞ η ( ξ − s ) + η U ( s ) ds, where ζ = ξ + iη . We remark that the Poisson integrals for the upper half plane andthe unit disk are related by the M¨obius transformation Φ( ζ ) = ( ζ − i ) / ( ζ + i ), moreprecisely, P U (Φ − ( w )) = P D U ◦ Φ − ( w ) holds (see [1, Chap.6] for basic properties of thePoisson integrals). Let I = ( σ, τ ) ( σ < τ ) be an interval and χ I its characteristic EFLECTION PRINCIPLE FOR LIGHTLIKE LINE SEGMENTS 5 function. Then we can easily see that the harmonic measure of I is given by P χ I ( ζ ) = 1 π { Arg( τ − ζ ) − Arg( σ − ζ ) } = 1 π (cid:8) Arg + ( τ − ζ ) − Arg + ( σ − ζ ) (cid:9) . Here, Arg and Arg + denote the harmonic blanches of arg defined on C \ ( −∞ , C \ [0 , ∞ ) which take values in ( − π, π ) and (0 , π ), respectively. Note that(2.1) Arg + ( − ζ ) = Arg( ζ ) + π holds for ζ ∈ C \ ( −∞ , Theorem 2.3 (Reflection principle for lightlike lines) . If a lightlike line segment L ⊂ ∂ Σ has shrinking singularities at its endpoints a, b ∈ L , then Σ extends to asurface with zero mean curvature across L .Further, the extended surface is invariant under the point symmetry with respectto the midpoint c = ( a + b ) / of L . In particular, it is maximal, except along L .Proof. Let X : H → Σ be an isothermal parametrization from the upper half plane.Then, X can be written as a Poisson integral of some bounded map b X : R ≃ ∂ H → L . In this case, the assumption that L has shrinking singularities at its endpoints a and b is translated into the fact that there exists s ∈ R such that b X ≡ a on aninterval ( s + σ, s ) and b X ≡ b on an interval ( s , s + τ ), where σ < < τ . Bycomposing a M¨obius transformation, we may assume s = 0.Let Π : D + := R > × (0 , π ) → H be a homeomorphism defined by Π( r, θ ) = re iθ .Observe that Π is real analytic on a domain D := R × (0 , π ) wider than D + , andΠ( D ) = H ∪ { } ∪ H ∗ , where H ∗ denotes the lower half plane. We prove that thereal analytic map X ◦ Π on D + extends to D real analytically. First, the linearityof the Poisson integral implies X ( ζ ) = P b X ( ζ ) = aP χ ( σ, ( ζ ) + bP χ (0 ,τ ) ( ζ ) + P W ( ζ )= aπ (cid:8) Arg + ( − ζ ) − Arg + ( σ − ζ ) (cid:9) + bπ { Arg( τ − ζ ) − Arg( − ζ ) } + P W ( ζ ) , where ζ = Π( r, θ ) and W = (1 − χ ( σ,τ ) ) b X . By (2.1), we have Arg + ( − ζ ) = Arg( re iθ )+ π = θ + π, and Arg( − ζ ) = Arg + ( re iθ ) − π = θ − π . Thus, X ( ζ ) = a + b + a − bπ θ + bπ Arg( τ − ζ ) − aπ Arg + ( σ − ζ ) + P W ( ζ ) . First three terms are clearly real analytic on D . Further since ( σ − Π( D )) ⊂ C \ [0 , + ∞ ) and ( τ − Π( D )) ⊂ C \ ( −∞ , D . Finally, observe that P W is harmonic on H , continuous on H ∪ ( σ, τ ) and P W ≡ (0 , , 0) on ( σ, τ ). Thus the Schwarz reflection principle impliesthat P W extends to H ∪ ( σ, τ ) ∪ H ∗ harmonically by P W ( ζ ) = − P W ( ζ ). SinceΠ( D ) ⊂ H ∪ ( σ, τ ) ∪ H ∗ , we conclude that P W ◦ Π and therefore X ◦ Π are realanalytic on D . Thus, Σ extends to a surface with zero mean curvature across L . S. AKAMINE AND H. FUJINO For the latter statement, note that Π( − r, π − θ ) = re − iθ = ζ for ζ = Π( r, θ ). Thus X ◦ Π( − r, π − θ ) = a + b + a − bπ ( π − θ )+ bπ Arg (cid:16) ( τ − ζ ) (cid:17) − aπ Arg + (cid:16) ( σ − ζ ) (cid:17) + P W ( ζ )= a + b + a − b − a − bπ θ + bπ ( − Arg( τ − ζ )) − aπ (cid:0) π − Arg + ( σ − ζ ) (cid:1) − P W ( ζ )= − a − bπ θ − bπ Arg( τ − ζ ) + aπ Arg + ( σ − ζ ) − P W ( ζ ) . We have X ◦ Π( − r, π − θ ) + X ◦ Π( r, θ ) = a + b . This implies the desired reflectionsymmetry. (cid:3) Figure 3. Blow-up of a discontinuous point of b X and the real ana-lytic extension X ◦ Π across a boundary lightlike line segment. Remark . In the above situation, it holds that X ◦ Π(0 , θ ) = b + a − bπ θ = a θπ + b (cid:18) − θπ (cid:19) . Therefore, X ◦ Π(0 , θ ) is the dividing point of L which divides L into two segmentswith Euclidean lengths (1 − θ/π ) : θ/π .2.4. Methods of finding shrinking singularities on lightlike lines. At theend of this section, we give some useful criteria for the endpoint of a lightlike linesegment boundary to be a shrinking singularity. By using this, we can find shrinkingsingularities on lightlike line segments from the shape of the surfaces.Let L ⊂ ∂ Σ be a lightlike line segment to which ψ tamely degenerates. Further,let X : D → Σ = graph( ψ ) be an isothermal parametrization, and b X : ∂ D → L itsboundary value function, that is, X ( w ) = P D b X ( w ). Then, as mentioned above, thereis a jump point w ∈ ∂ D of b X such that L ⊂ C ( X, w ). If b X maps some arc of oneside of w constantly to an endpoint of L , we say that the endpoint is a shrinkingsingularity . Then, L has shrinking singularities at its endpoints if and only if bothof the two endpoints are shrinking singularities.One of the most typical situations where shrinking singularities appear on lightlikeline segments is as follows. EFLECTION PRINCIPLE FOR LIGHTLIKE LINE SEGMENTS 7 Fact 2.5 ([3, Theorem 4.5]) . Assume that ∂ Σ contains two adjacent lightlike linesegments L and L whose union does not form a straight line segment. If ψ tamelydegenerates to L and L , respectively, then the common endpoint of L and L isa shrinking singularity. More precisely, Fact 2.5 says that if we let w , w ∈ ∂ D be corresponding jumppoints to L and L , respectively, then w = w and one of the two arcs of ∂ D joining w and w is constantly mapped to the common endpoint of L and L by b X , see Figure 4. Figure 4. A situation to which Fact 2.5 is applicable.We shall remark that the same is true even if Ω has a slit (see Figure 5). Proposition 2.6. Let I ⊂ Ω be an open Euclidean segment joining an interior point p ∈ Ω and a boundary point. Assume that a maximal graph Σ = graph( ψ ) over adomain Ω \ I contains two lightlike line segments L and L (possibly coincidingwith each other) in its boundary over the slit I . If ψ tamely degenerates to L and L , respectively, then the common endpoint of L and L over p is a shrinkingsingularity. Figure 5. Situations to which Proposition 2.6 are applicable. Proof. The proof is completely the same as [3, Theorem 4.5]. To do this, we onlyneed to check that the Hengartner-Schober theorem [16, Theorem 4.3], [3, Lemma3.4] can be applied to Ω \ I . However, in fact, this theorem can be applied to anybounded simply connected domain with locally connected boundary as stated intheir original paper [16, Theorem 4.3] (see also [7, Section 3.3]). (cid:3) We explain how to use Fact 2.5 and Proposition 2.6 with examples below: S. AKAMINE AND H. FUJINO Example . The first example is a singly periodic maximal surface of Riemanntype illustrated on the left side of Figure 1, which is given by S = { ( x, y, t ) ∈ L | − x + t ) sin t − ( x + y − xt + t ) cos t = 0 } , (see [2, Theorem 5.3 (1-i)]). Choose one sheet of S as on the left of Figure 6. Thenthe sheet is an entire graph which is maximal except at two points that are isolatedsingularities and the lightlike line segment joining them. If we restrict the graph toa domain Ω like in the right side of Figure 6, then the restricted maximal graphtamely degenerates to the lightlike line segment from each side, since it extendsto the entire graph. Thus Proposition 2.6 implies that the lightlike line segmenthas shrinking singularities at its endpoints. Consequently, S is invariant under thepoint symmetry at the midpoint of the lightlike line segment by Theorem 2.3. Notethat maximal surfaces of Riemann type without lightlike lines were determined viathe Weierstrass representation by L´opez-L´opez-Souam [23]. Figure 6. One sheet of S (left) and a restricted graph (right).The second and third examples are the following doubly and triply periodic max-imal surfaces illustrated in the center and on the right of Figure 1, respectively: S = { ( x, y, t ) ∈ L | cos t cosh y + cos x = 0 } , S = { ( x, y, t ) ∈ L | cos t − cos x cos y = 0 } . For instance, we similarly take one sheet of S as on the left of Figure 7. Then thesheet is an entire maximal graph with isolated singularities and lightlike line seg-ments. In this case, for each isolated singularity there are two lightlike line segmentsjoining it, and thus Fact 2.5 is applicable to the graph restricted to a domain Ω as on the right side of Figure 7, by the same argument as for S . Therefore, eachlightlike line segment has shrinking singularities at its endpoints, and S is invariantunder the point symmetry about this midpoint.The same is true for the surface S . Note that S is called the spacelike Scherksurface in [13]. For each shrinking singularity on this surface there are four lightlikelines passing through it (see Figure 1, right). Further, this surface also can be con-structed by the method described in Section 3 (apply the tessellation generated by asquare). We also remark that periodic maximal surfaces with shrinking singularitiesbut without lightlike line segments were intensively studied in [10, 12, 23] (see alsotheir references). EFLECTION PRINCIPLE FOR LIGHTLIKE LINE SEGMENTS 9 Figure 7. One sheet of S (left) and a restricted graph (right).3. Construction of periodic maximal surfaces with lightlike lines The reflection principle in Theorem 2.3 gives a new technique to extend and con-struct maximal surfaces with lightlike line segments. As an application of Theorem2.3, we construct triply periodic maximal surfaces with lightlike lines.3.1. Procedures of construction. The procedures are as follows: Step 1 . Give a tessellation of R which is made by a polygon Ω satisfying thefollowing two conditions (see Figure 8, top left):(1-a) there exists a maximal graph Σ which tamely degenerates to future andpast-directed lightlike line segments on the edges of Ω alternately.(1-b) the tessellation is generated by point-symmetries of Ω with respect to itsmidpoints of edges. Step 2 . Construct a doubly periodic entire maximal graph Σ from Σ by iteratingthe reflection principle for lightlike line segments. The period lattice Λ correspondsto that of the tessellation in Step (1-b) (see Figure 8, top right). Step 3 . Construct a triply periodic proper maximal surface Σ from Σ by iteratingthe reflection principle for shrinking singularities. The period lattice corresponds tobasis of Λ and a lightlike vector along a lightlike line segment (see Figure 8, bottom).3.2. Details and examples.Step (1-a) : One way to construct maximal graphs with lightlike line boundariesin Step (1-a) is to use Jenkins and Serrin’s criteria [18] for infinite boundary valueproblems of the minimal surface equation in the Euclidean 3-space:Let Ω ⊂ R be a polygonal domain whose boundary consists of a finite numberof open line segments A , . . . , A k , B , . . . , B l . For each of families { A j } and { B j } ,assume that no two of the elements meet to form a convex corner. Further, fora polygonal domain P ⊂ Ω whose vertices are those of Ω, let α P and β P denoterespectively, the total length of A j such that A j ⊂ ∂P and the total length of B j such that B j ⊂ ∂P , and let γ P be the perimeter of P .Then, by the duality of boundary value problems for minimal surfaces in E and maximal surfaces in L proved in [3, Theorem 1], the classical Jenkins-Serrin’stheorem [18, Theorem 3] yields the following result. Fact 3.1. There exists a maximal graph Σ over Ω which tamely degenerates toa future-directed lightlike line segment on each A j and a past-directed lightlike line Figure 8. The surfaces Σ in Step 1 (top left), Σ in Step 2 (topright) and Σ in Step 3 (bottom). segment on each B j if and only if (3.1) 2 α P < γ P and β P < γ P hold for each polygonal domain P ( Ω taken as above and (3.2) α Ω = β Ω holds. The solution is unique up to an additive constant if it exists.Remark . Each maximal graph over a polygonal domain Ω in Fact 3.1 can beconstructed by using the Poisson integral of a step function on S ≃ ∂ D valued inthe vertices of Ω, which is a good way to construct maximal graphs explicitly. See[3, Corollary 3.12 and Section 4.4] for more details. Step (1-b) : Let M be the set of polygonal domains which tessellate R by takingpoint symmetries with respect to the midpoints of all edges repeatedly.The following assertions give a classification of the shapes of domains in M sat-isfying the conditions in Fact 3.1. See Figure 9. Lemma 3.3. Each Ω ∈ M is either one of the following: (i) a triangle, (ii) a quadrilateral, or (iii) a hexagon whose opposite sides are parallel and of equal length. Lemma 3.4. A domain Ω ∈ M satisfying (3.1) and (3.2) is either (ii ′ ) a quadrilateral whose two pairs of opposite sides are equal in length, or (iii ′ ) a hexagon satisfying the conditions (iii) in Lemma . and (3.3) d > | b − ( a + c ) | , where a, b, c are the lengths of consecutive three edges and d is the smallestlength of diagonal lines connecting opposite vertices of the hexagon. EFLECTION PRINCIPLE FOR LIGHTLIKE LINE SEGMENTS 11 Figure 9. Tessellations of R from polygonal domains in M .The proofs are given in Appendix A. Therefore, we can construct the maximal graphΣ over a given polygonal domain Ω in Lemma 3.4 (see Figure 10). Figure 10. Maximal graphs with lightlike line boundaries overpolygonal domains in M generated by Step 1. Step 2 : By Fact 2.5, the endpoints of each lightlike line segment are shrinkingsingularities. Applying Theorem 2.3, i.e. taking point symmetries at the midpointsof the lightlike line segments, we have a doubly periodic maximal graph Σ withshrinking singularities on the vertices and lightlike line segments over the edges ofthe polygons (see Figure 11). Figure 11. Doubly periodic entire maximal graphs with lightlikeline segments and shrinking singularities generated by Step 2. Step 3 : Applying the reflection principle for shrinking singularities, i.e. taking pointsymmetries there, we have a triply periodic proper maximal surface Σ , which is amulti-valued graph with infinitely many sheets congruent to Σ (see Figure 12). Figure 12. Triply periodic maximal surfaces with lightlike line seg-ments and shrinking singularities generated by Step 3.3.3. Parametric representation of maximal surfaces with lightlike lines. Well-known classes of maximal surfaces with singularities such as generalized max-imal surfaces in [8] and maxfaces in [26] are defined on Riemann surfaces. Un-fortunately, isothermal coordinates break down near lightlike line segments for theperiodic maximal surfaces Σ and Σ in the previous subsection. Therefore, toparametrize them we need to consider a wider class of maximal surfaces. Definition 3.5 ([26, Definition 2.1]) . A smooth map X : M → L from a 2-dimensional manifold M to L is called a maximal map if there is an open dense set W ⊂ M such that X | W is a spacelike maximal immersion.Suppose the boundary of a maximal graph contains a lightlike line segment withshrinking singularities at its endpoints. Then, as shown in the proof of Theorem2.3, the extended surface across the lightlike line segment can be parametrized bya non-conformal but real analytic mapping (see Figure 3). Moreover, the extensionwith respect to each of the shrinking singularities is parametrized by a general-ized maximal surface, in particular, an analytic mapping. As a consequence, forexample, the triply periodic maximal surfaces constructed in Section 3.1 can beparametrized by a (non-conformal) real analytic proper maximal map by gluing theabove parametrizations together. In this case, the domain of the maximal map is a2-dimensional manifold of infinite genus. See Figure 13.4. Concluding remarks and a future problem Finally, we remark that the reflection principle in Theorem 2.3 is valid only forlightlike line segments connecting shrinking singularities. Therefore, it is natural toconsider the following cases, as well: The boundary of a maximal graph contains Case 1 an entire lightlike line, or Case 2 a lightlike half-line.On the contrary, Theorem 2.3 is not valid for such entire lightlike lines or lightlikehalf-lines, in particular, we cannot take the “midpoint” of any such lines. See Figure14, for example. Typical examples of Case 2 are the following hyperbolic catenoid H and the parabolic catenoid P , which are invariant under rotations in L with respect EFLECTION PRINCIPLE FOR LIGHTLIKE LINE SEGMENTS 13 Figure 13. The surface Σ parametrized by a proper maximal map.to spacelike and lightlike axes, respectively (see Figure 14 and also [13, 14] for theirimplicit representations): H = { ( x, y, t ) ∈ L | sin x + y − t = 0 } , P = { ( x, y, t ) ∈ L | x − t ) − ( x + t ) + 12 y = 0 } . Also, examples of Case 1 were given in [4], and it is known that such a maximal Figure 14. The hyperbolic catenoid (left) and the paraboliccatenoid (right).graph containing an entire lightlike line cannot be defined on a convex domain, asproved in [9, Lemma 2.1].From the viewpoint of the function theory, in the present article we discussedbounded harmonic mappings. On the other hand, Cases 1 and 2 lead us to consid-ering unbounded harmonic mappings for which the Poisson integrals do not work. Related to this, the following problem remains as a future work. Question 2. Is there a reflection principle for entire lightlike lines or lightlike half-lines (as in the above Cases and ) on the boundaries of maximal surfaces? Appendix A. Proofs of Lemmas 3.3 and 3.4 In this appendix, we give a classification of tessellations of R satisfying the con-ditions of Step 1 in Section 3.1.For a chosen vertex z of a polygonal domain Ω, we denote the interior angles of Ωstarting from z by α , α , . . . , α n and set α mn + k = α k for a positive integer m and k = 1 , , . . . , n . Lemma A.1. If Ω ∈ M , then there exists a unique j ≥ such that α + α + · · · + α j = 2 π and α l = α m if l ≡ k mod j . Moreover, this j is independent of a choice ofvertices of Ω . We call the above j for Ω ∈ M the valency of Ω, and in this case each vertex issaid to be j-valent . Proof. Let us consider the edge connecting the vertices with the angles α and α .If we take the reflection with respect to the midpoint of the edge, then α appearsnext to α around z . Inductively, α , α , α , . . . appear around z in this order (seeFigure 15). Thus there is some j such that α + α + · · · + α j = 2 π and α l = α m if l ≡ k mod j .Next, we denote the valencies of the vertices with angles α and α by j and j .Then α + α + · · · + α j = 2 π and α j +1 = α hold. Hence we have α + α + · · · + α j +1 = α + α + · · · + α j = 2 π, which implies j = j . By induction on the vertices, we obtain the last assertion. (cid:3) Figure 15. The condition α + α + · · · + α j = 2 π at the vertex z . Lemma A.2. Let Ω ∈ M be an n -gon and each vertex is j -valent for j ≥ . Thenthe following equation holds. ( n − j = 2 n. In particular, the possible values of n and j are ( n, j ) = (3 , , (4 , , (6 , . EFLECTION PRINCIPLE FOR LIGHTLIKE LINE SEGMENTS 15 Proof. By Lemma A.1, the relation α i + α i +1 + · · · + α i + j − = 2 π and α i + j = α i hold for arbitrary i . Since the sum of the interior angles of an n -gon is ( n − π , weobtain j ( n − π = j ( α + α + · · · + α n )= n X i =1 ( α i + α i +1 + · · · + α i + j − )= 2 nπ, which is the desired equality. (cid:3) It can be easily seen that any triangles and quadrilaterals are in M , and they are6-valent and 4-valent, respectively. When Ω ∈ M is a hexagon, which is 3-valent,Lemma A.1 yields that the interior angles of Ω are written as α , α , α , α , α , α in this order. This implies that the opposite sides of edges of Ω are parallel and ofequal length, and hence we have a proof of Lemma 3.3.Finally, the proof of Lemma 3.4 is completed as follows. Proof of Lemma . . It is enough to consider the three cases in Lemma A.2. Anytriangle does not satisfy (3.2) by the triangle inequality. 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