PPERIODIC MINIMAL SURFACES INSEMIDIRECT PRODUCTS
ANA MENEZES
Abstract
In this paper we prove existence of complete minimal surfaces in somemetric semidirect products. These surfaces are similar to the doubly andsingly periodic Scherk minimal surfaces in R . In particular, we obtainthese surfaces in the Heisenberg space with its canonical metric, and inSol with a one-parameter family of non-isometric metrics. Mathematics Subject Classification (2010):
Key words:
Semidirect products; minimal surfaces.
In this paper we construct examples of periodic minimal surfaces in somesemidirect products R (cid:111) A R , depending on the matrix A. By periodic surfacewe mean a properly embedded surface invariant by a nontrivial discrete groupof isometries.One of the most simple examples of semidirect product is H × R = R (cid:111) A R , when we take A = (cid:18) (cid:19) . In this space, Mazet, Rodr´ıguez and Rosenberg[2] proved some results about periodic constant mean curvature surfaces andthey constructed examples of such surfaces. One of their methods is to solvea Plateau problem for a certain contour. In [5], using a similar technique,Rosenberg constructed examples of complete minimal surfaces in M × R , where M is either the two-sphere or a complete Riemannian surface withnonnegative curvature or the hyperbolic plane.Meeks, Mira, P´erez and Ros [3] have proved results concerning the ge-ometry of solutions to Plateau type problems in metric semidirect products R (cid:111) A R , when there is some geometric constraint on the boundary values ofthe solution (see Theorem 1).The first example that we construct is a complete periodic minimal surfacesimilar to the doubly periodic Scherk minimal surface in R . It is invariant bytwo translations that commute and is a four punctured sphere in the quotientof R (cid:111) A R by the group of isometries generated by the two translations. In1 a r X i v : . [ m a t h . DG ] S e p A. MENEZES the last section we obtain a complete periodic minimal surface analogous tothe singly periodic Scherk minimal surface in R . These surfaces are obtained by solving the Plateau problem for a geodesicpolygonal contour Γ (it uses a result by Meeks, Mira, P´erez and Ros [3] aboutthe geometry of solutions to the Plateau problem in semidirect products), andletting some sides of Γ tend to infinity in length, so that the associated Plateausolutions all pass through a fixed compact region (this will be assured by theexistence of minimal annuli playing the role of barriers). Then a subsequenceof the Plateau solutions will converge to a minimal surface bounded by ageodesic polygon with edges of infinite length. We complete this surface bysymmetry across the edges. The whole construction requires precise geometriccontrol and uses curvature estimates for stable minimal surfaces.These results are obtained for semidirect products R (cid:111) A R where A = (cid:18) bc (cid:19) . For example, we obtain periodic minimal surfaces in the Heisenbergspace, when A = (cid:18) (cid:19) , and in Sol , when A = (cid:18) (cid:19) , with their wellknown Riemannian metrics. When we consider the one-parameter family ofmatrices A ( c ) = (cid:18) c c (cid:19) , c ≥ , we get a one-parameter family of metricsin Sol which are not isometric. Acknowledgements.
This work is part of the author’s Ph.D. thesis at IMPA. Theauthor would like to express her sincere gratitude to her advisor Prof. Harold Rosen-berg for his constant encouragement and guidance throughout the preparation of thiswork. The author would also like to thank Joaqu´ın P´erez for helpful conversationsabout semidirect products. The author was financially suported by CNPq-Brazil andIMPA.
Generalizing direct products, a semidirect product is a particular way in whicha group can be constructed from two subgroups, one of which is a normalsubgroup. As a set, it is the cartesian product of the two subgroups but witha particular multiplication operation.In our case, the normal subgroup is R and the other subgroup is R . Givena matrix A ∈ M ( R ) , we can consider the semidirect product R (cid:111) A R , wherethe group operation is given by( p , z ) ∗ ( p , z ) = ( p + e z A p , z + z ) , p , p ∈ R , z , z ∈ R (2.1)and A = (cid:18) a bc d (cid:19) ∈ M ( R ) . We choose coordinates ( x, y ) ∈ R , z ∈ R . Then ∂ x = ∂∂x , ∂ y , ∂ z is a par-allelization of G = R (cid:111) A R . Taking derivatives at t = 0 in (2.1) of the left ERIODIC MINIMAL SURFACES IN SEMIDIRECT PRODUCTS t, , ∈ G (respectively by (0 , t, , (0 , , t )), we obtain thefollowing basis { F , F , F } of the right invariant vector fields on G : F = ∂ x , F = ∂ y , F = ( ax + by ) ∂ x + ( cx + dy ) ∂ y + ∂ z . (2.2)Analogously, if we take derivatives at t = 0 in (2.1) of the right multiplica-tion by ( t, , ∈ G (respectively by (0 , t, , (0 , , t )), we obtain the followingbasis { E , E , E } of the Lie algebra of G : E = a ( z ) ∂ x + a ( z ) ∂ y , E = a ( z ) ∂ x + a ( z ) ∂ y , E = ∂ z , (2.3)where we have denoted e zA = (cid:18) a ( z ) a ( z ) a ( z ) a ( z ) (cid:19) . We define the canonical left invariant metric on R (cid:111) A R , denoted by (cid:104) , (cid:105) , to be that one for which the left invariant basis { E , E , E } is orthonormal.The expression of the Riemannian connection ∇ for the canonical left in-variant metric of R (cid:111) A R in this frame is the following: ∇ E E = aE ∇ E E = b + c E ∇ E E = − aE − b + c E ∇ E E = b + c E ∇ E E = dE ∇ E E = − b + c E − dE ∇ E E = c − b E ∇ E E = b − c E ∇ E E = 0 . In particular, for every ( x , y ) ∈ R , γ ( z ) = ( x , y , z ) is a geodesic in G. Remark 1. As [ E , E ] = 0 , thus for all z, R (cid:111) A { z } is flat and the horizontalstraight lines are geodesics. Moreover, the mean curvature of R (cid:111) A { z } withrespect to the unit normal vector field E is the constant H = tr ( A ) / . The change from the orthonormal basis { E , E , E } to the basis { ∂ x , ∂ y , ∂ z } produces the following expression for the metric (cid:104) , (cid:105) : (cid:104) , (cid:105) ( x,y,z ) = [ a ( − z ) + a ( − z ) ] dx + [ a ( − z ) + a ( − z ) ] dy + dz +[ a ( − z ) a ( − z ) + a ( − z ) a ( − z )]( dx ⊗ dy + dy ⊗ dx )= e − tr ( A ) z { [ a ( z ) + a ( z ) ] dx + [ a ( z ) + a ( z ) ] dy } + dz − e − tr ( A ) z [ a ( z ) a ( z ) + a ( z ) a ( z )]( dx ⊗ dy + dy ⊗ dx ) . In particular, for every matrix A ∈ M ( R ) , the rotation by angle π aroundthe vertical geodesic γ ( z ) = ( x , y , z ) given by the map R ( x, y, z ) = ( − x +2 x , − y + 2 y , z ) is an isometry of ( R (cid:111) A R , (cid:104) , (cid:105) ) into itself. A. MENEZES
Remark 2.
As we observed, the vertical lines of R (cid:111) A R are geodesics ofits canonical metric. For any line l in R (cid:111) A { } let P l denote the verticalplane { ( x, y, z ) : ( x, y, ∈ l ; z ∈ R } containing the set of vertical lines passingthrough l. It follows that P l is ruled by vertical geodesics and, since rotation byangle π around any vertical line in P l is an isometry that leaves P l invariant,then P l has zero mean curvature. Although the rotation by angle π around horizontal geodesics is not alwaysan isometry, we have the following result. Proposition 1.
Let A = (cid:18) bc (cid:19) ∈ M ( R ) and consider the horizontalgeodesic α = { ( x , t,
0) : t ∈ R } in R (cid:111) A { } parallel to the y -axis. Then therotation by angle π around α is an isometry of ( R (cid:111) A R , (cid:104) , (cid:105) ) into itself. Thesame result is true for a horizontal geodesic parallel to the x -axis.Proof. The rotation by angle π around α is given by the map φ ( x, y, z ) =( − x + 2 x , y, − z ) , so φ x = − ∂ x , φ y = ∂ y and φ z = − ∂ z . If A = (cid:18) bc (cid:19) , thene zA = ∞ (cid:88) k =0 ( bc ) k z k (2 k )! ∞ (cid:88) k =1 b k c k − z k − (2 k − ∞ (cid:88) k =1 c k b k − z k − (2 k − ∞ (cid:88) k =0 ( bc ) k z k (2 k )! . Hence, a ( z ) = a ( z ) and e − zA = (cid:18) a ( z ) − a ( z ) − a ( z ) a ( z ) (cid:19) . Then (cid:104) , (cid:105) ( x,y,z ) = { [ a ( z ) + a ( z ) ] dx + [ a ( z ) + a ( z ) ] dy } + dz − [ a ( z ) a ( z ) + a ( z ) a ( z )]( dx ⊗ dy + dy ⊗ dx )and (cid:104) , (cid:105) φ ( x,y,z ) = { [ a ( z ) + a ( z ) ] dx + [ a ( z ) + a ( z ) ] dy } + dz +[ a ( z ) a ( z ) + a ( z ) a ( z )]( dx ⊗ dy + dy ⊗ dx ) . Therefore, (cid:104) φ x , φ x (cid:105) φ ( x,y,z ) = (cid:104) ∂ x , ∂ x (cid:105) ( x,y,z ) , (cid:104) φ y , φ y (cid:105) = (cid:104) ∂ y , ∂ y (cid:105) , (cid:104) φ z , φ z (cid:105) = (cid:104) ∂ z , ∂ z (cid:105) , that is, φ is an isometry. Analogously, we can show that the rota-tion by angle π around a horizontal geodesic parallel to the x -axis is also anisometry. Remark 3.
When the matrix A in R (cid:111) A R is (cid:18) (cid:19) and (cid:18) (cid:19) , we have the Heisenberg space and Sol , respectively, with their well known ERIODIC MINIMAL SURFACES IN SEMIDIRECT PRODUCTS Riemannian metrics. When we consider the one-parameter family of matrices A ( c ) = (cid:18) c c (cid:19) , c ≥ , we get a one-parameter family of metrics in Sol which are not isometric. For more details, see [4]. Meeks, Mira, P´erez and Ros [3] have proved results concerning the ge-ometry of solutions to Plateau type problems in metric semidirect products R (cid:111) A R , when there is some geometric constraint on the boundary values ofthe solution. More precisely, they proved the following theorem. Theorem 1 (Meeks, Mira, P´erez and Ros, [3]) . Let X = R (cid:111) A R be a metricsemidirect product with its canonical metric and let Π : R (cid:111) A R → R (cid:111) A { } denote the projection Π( x, y, z ) = ( x, y, . Suppose E is a compact convex diskin R (cid:111) A { } , C = ∂E and Γ ⊂ Π − ( C ) is a continuous simple closed curvesuch that Π : Γ → C monotonically parametrizes C. Then,1. Γ is the boundary of a compact embedded disk Σ of finite least area.2. The interior of Σ is a smooth Π -graph over the interior of E. Throughout this section, we consider the semidirect product R (cid:111) A R withthe canonical left invariant metric (cid:104) , (cid:105) , where A = (cid:18) bc (cid:19) . In this space,we prove the existence of a complete minimal surface analogous to Scherk’sdoubly periodic minimal surface in R .Fix 0 < c < c and let a be a sufficiently small positive quantity such that a < (cid:90) c c (cid:113) a ( z ) + a ( z ) + (cid:113) a ( z ) + a ( z )d z − (cid:90) c c (cid:112) ( a ( z ) + a ( z )) + ( a ( z ) + a ( z )) d z. (3.1)Note that such positive number a exists, as | ∂ x | = (cid:112) a ( z ) + a ( z ) , | ∂ y | = (cid:112) a ( z ) + a ( z ) and | ∂ x + ∂ y | = (cid:112) ( a ( z ) + a ( z )) + ( a ( z ) + a ( z )) . For each c > , consider the polygon P c in R (cid:111) A R with the sides α , α , α c , α c and α c defined below. α = { ( t, ,
0) : 0 ≤ t ≤ a } α = { (0 , t,
0) : 0 ≤ t ≤ a } α c = { ( a, , t ) : 0 ≤ t ≤ c } α c = { (0 , a, t ) : 0 ≤ t ≤ c } α c = { ( t, − t + a, c ) : 0 ≤ t ≤ a } , A. MENEZES as illustrated in Figure 1. Figure 1: Polygon P c . We will denote α = { ( t, ,
0) : 0 ≤ t < a } , α = { (0 , t,
0) : 0 ≤ t < a } ,α = { ( a, , t ) : t > } and α = { (0 , a, t ) : t > } , hence P ∞ = α ∪ α ∪ α ∪ α ∪ { ( a, , , (0 , a, } . Let Π : R (cid:111) A R → R (cid:111) A { } denote the projection Π( x, y, z ) = ( x, y, . The next proposition is proved in Lemma 1.2 in [3], using the maximum prin-ciple and the fact that for every line L ⊂ R (cid:111) A { } , the vertical plane Π − ( L )is a minimal surface. Proposition 2.
Let E be a compact convex disk in R (cid:111) A { } with boundary C = ∂E and let Σ be a compact minimal surface with boundary in Π − ( C ) . Then every point in intΣ is contained in intΠ − ( E ) . Observe that, for each c > , the polygon P c is transverse to the Killingfield X = ∂ x + ∂ y and each integral curve of X intersects P c in at most onepoint. From now on, denote by P the commom projection of every P c over R (cid:111) A { } , that is, P = Π( P c ) = Π( P d ) for any c, d ∈ R , and denote by E thedisk in R (cid:111) A { } with boundary P. Let us denote by R the region E ×{ z ≥ } . Using Theorem 1, we conclude that P c is the boundary of a compact embeddeddisk Σ c of finite least area and the interior of Σ c is a smooth Π-graph over theinterior of E. Let Ω c = { ( t, − t + a, s ) : 0 ≤ t ≤ a ; 0 ≤ s ≤ c } . Proposition 3. If S is a compact minimal surface with boundary P c , then S = Σ c . Proof.
By Proposition 2, intΣ c , int S ⊂ intΠ − ( E ) , then, in particular, intΣ c , int S ⊂ int { ϕ t ( p ) : t ∈ R ; p ∈ Ω c } , where ϕ t is the flow of the Killing field X. As S is compact, there exists t such that ϕ t (Σ c ) ∩ S = ∅ . If S (cid:54) = Σ c , thenthere exists t > ϕ t (Σ c ) ∩ S (cid:54) = ∅ and for t > t , ϕ t (Σ c ) ∩ S = ∅ . Since for all t (cid:54) = 0 , ϕ t ( P c ) ∩ S = ∅ , then the point of intersection is an interiorpoint and, by the maximum principle, ϕ t (Σ c ) = S. But that is a contradiction,since t (cid:54) = 0 . Therefore, S = Σ c . ERIODIC MINIMAL SURFACES IN SEMIDIRECT PRODUCTS
Proposition 4.
Let N be a homogeneous three-manifold. Let Σ n be an ori-ented properly embedded minimal surface in N. Suppose there exist c > such that for all n, | A Σ n | ≤ c, and a sequence of points { p n } in Σ n such that p n → p ∈ N. Then there exists a subsequence of Σ n that converges to a com-plete minimal surface Σ with p ∈ Σ . Here A Σ n denotes the second fundamentalform of Σ n . For each n ∈ N , let Σ n be the solution to the Plateau problem with bound-ary P n . By Theorem 1 and Proposition 3, Σ n is stable and unique. We areinterested in proving the existence of a subsequence of Σ n that convergesto a complete minimal surface with boundary P ∞ . In order to do that, wewill use a minimal annulus as a barrier (whose existence is guaranteed bythe Douglas criterion (see [1], Theorem 2.1)) to show that there exist points p n ∈ Σ n , Π( p n ) = q ∈ int E for all n, which converge to a point p ∈ R (cid:111) A R , and then we will use Proposition 4.Consider the parallelepiped with the faces A, B, C, D, E and F , definedbelow. A = { ( u, − (cid:15), v ) : (cid:15) ≤ u ≤ a + (cid:15) ; c ≤ v ≤ c } B = { ( − (cid:15), u, v ) : (cid:15) ≤ u ≤ a + (cid:15) ; c ≤ v ≤ c } C = { ( u, − u, v ) : − (cid:15) ≤ u ≤ (cid:15) ; c ≤ v ≤ c } D = { ( u, − u + a, v ) : − (cid:15) ≤ u ≤ a + (cid:15) ; c ≤ v ≤ c } E = { ( u, − u + v, c ) : − (cid:15) ≤ u ≤ v + (cid:15) ; 0 ≤ v ≤ a } F = { ( u, − u + v, c ) : − (cid:15) ≤ u ≤ v + (cid:15) ; 0 ≤ v ≤ a } , where (cid:15) is a positive constant that we will choose later. Observe that eachone of these faces is the least area minimal surface with its boundary. Let usanalyse the area of each face.1. In the plane { y = constant } the induced metric is given by g ( x, z ) =( a ( z ) + a ( z )) dx + dz . Hence,area A = (cid:90) c c (cid:90) a + (cid:15)(cid:15) (cid:113) a ( z ) + a ( z )d x d z = a (cid:90) c c (cid:113) a ( z ) + a ( z )d z.
2. In the plane { x = constant } the induced metric is given by g ( y, z ) = A. MENEZES ( a ( z ) + a ( z )) dy + dz . Hence,area B = (cid:90) c c (cid:90) a + (cid:15)(cid:15) (cid:113) a ( z ) + a ( z )d x d z = a (cid:90) c c (cid:113) a ( z ) + a ( z )d z.
3. The face C is contained in the plane parameterized by φ ( u, v ) =( u, − u, v ) and the face D is contained in the plane parameterized by ψ ( u, v ) =( u, − u + a, v ). We have ψ u = φ u = ∂ x − ∂ y , ψ v = φ v = ∂ z . Then, | ψ u ∧ ψ v | = | φ u ∧ φ v | = (cid:112) ( a ( z ) + a ( z )) + ( a ( z ) + a ( z )) . Hence,area C = (cid:90) c c (cid:90) + (cid:15) − (cid:15) (cid:112) ( a ( z ) + a ( z )) + ( a ( z ) + a ( z )) d u d v = 2 (cid:15) (cid:90) c c (cid:112) ( a ( z ) + a ( z )) + ( a ( z ) + a ( z ) d z, area D = (cid:90) c c (cid:90) a + (cid:15) − (cid:15) (cid:112) ( a ( z ) + a ( z )) + ( a ( z ) + a ( z )) d u d v = ( a + 2 (cid:15) ) (cid:90) c c (cid:112) ( a ( z ) + a ( z )) + ( a ( z ) + a ( z )) d z.
4. As the plane { z = constant } is flat, then the induced metric is theEuclidean metric. Hence,area E = area F = (cid:90) a (cid:90) v + (cid:15) − (cid:15) d u d v = a ( a + 4 (cid:15) )2 . Therefore,area C + area D + area E + area F < area A + area B se, e somente se,( a + 4 (cid:15) ) (cid:20) a + (cid:90) c c (cid:112) ( a + a ) + ( a + a ) d z (cid:21) < a (cid:90) c c (cid:113) a + a d z + a (cid:90) c c (cid:113) a + a d z se, e somente se, (cid:15) < a (cid:90) c c (cid:113) a ( z ) + a ( z ) + (cid:113) a ( z ) + a ( z )d za + (cid:90) c c (cid:112) ( a ( z ) + a ( z )) + ( a ( z ) + a ( z )) d z − a . (3.2) ERIODIC MINIMAL SURFACES IN SEMIDIRECT PRODUCTS A .As we chose a satisfying (3 . , the factor in the right hand side of (3.2) isa positive number, then we can choose (cid:15) > A with boundary ∂A ∪ ∂B such that its projection Π( A ) contains points of int E, where E is the disk in R (cid:111) A { } with boundary P. (See Figure 2).As R (cid:111) A { z } is a minimal surface, the maximum principle implies that,for each c, Σ c is contained in the slab bounded by the planes { z = 0 } and { z = c } . Then for c < c , Σ c ∩ A = ∅ . As Σ c is unique, Σ c varies continuouslywith c, and when c increases the boundary ∂ Σ c = P c does not touch ∂ A . Therefore, using the maximum principle, Σ c ∩ A = ∅ for all c, and Σ c is underthe annulus A , which means that over any vertical line that intersects A andΣ c , the points of Σ c are under the points of A . Consider ϕ t the flow of the Killing field X = ∂ x + ∂ y . Observe that { ϕ t ( A ) } t< forms a barrier for all points p n ∈ Σ n such that Π( p n ) is con-tained in a neighborhood U ⊂ E of the origin o = (0 , , . Moreover, forany c < c we can use the flow ϕ t of the Killing field X and the maximumprinciple to conclude that Σ c is under Σ c in the same sense as before.As, by Theorem 1, each Σ n is a vertical graph in the interior, then Σ n ∩ Π − ( q ) is only one point p n , for every point q ∈ int E. Moreover, by the previousparagraph, the sequence p n = Σ n ∩ Π − ( q ) is monotone. Then, since we havea barrier, the sequence { p n = Σ n ∩ Π − ( q ) } converges to a point p ∈ Π − ( q ) , for all q ∈ U . In order to understand the convergence of the surfaces Σ n we need toobserve some properties of these surfaces.First, notice that, rotation by angle π around α , that we will denote by R α , is an isometry. By the Schwarz reflection, we obtain a minimal surface (cid:101) Σ n = Σ n ∪ R α (Σ n ) that has int α in its interior. Note that the boundary0 A. MENEZES of (cid:101) Σ n is transverse to the Killing field X = ∂ x + ∂ y , and if ϕ t denotes theflow of X, we have that ϕ t ( ∂ (cid:101) Σ n ) ∩ (cid:101) Σ n = ∅ for all t (cid:54) = 0 , hence, using thesame arguments of the proof of Proposition 3, we can show that the minimalsurface (cid:101) Σ n is the unique minimal surface with its boundary. In particular, itis area-minimizing, and then it is stable. Hence, by Main Theorem in [6], wehave uniform curvature estimates for points far from the boundary of (cid:101) Σ n . Inparticular, we get uniform curvature estimates for Σ n in a neighborhood of α . Analogously, we have uniform curvature estimates for Σ n in a neighborhoodof α . Hence, for every compact contained in { z > } ∩ R , there exists a subse-quence of Σ n that converges to a minimal surface. Taking an exhaustion bycompact sets and using a diagonal process, we conclude that there exists asubsequence of Σ n that converges to a minimal surface Σ that has α ∪ α inits boundary. From now on, we will use the notation Σ n for this subsequence.It remains to prove that in fact Σ is a minimal surface with boundary P ∞ . In order to do that, we will use the fact that the interior of each Σ n is a verticalgraph over the interior of E . Let us denote by u n the function defined in int E such that Σ n = Graph( u n ) . We already know that u n − < u n in int E for all n. Claim 1.
There are uniform gradient estimates for { u n } for points in α ∪ α . Proof.
For x < δ > β = { ( x , y, c ) : − δ ≤ y ≤ } , β = { ( x , t, − c a t + c ) : 0 ≤ t ≤ a } , β = { ( x , t − δ, − c a t + c ) : 0 ≤ t ≤ a } and β = { ( x , y,
0) : a − δ ≤ y ≤ a } . This is a minimal surface transversal to the Killing field ∂ x , hence any smallperturbation of its boundary gives a minimal surface with that perturbedboundary. Thus, if we consider a small perturbation of the boundary of thisvertical strip by just perturbing a little bit β by a curve contained in { x ≥ x } joining the points ( x , − δ, c ) and ( x , , c ) , we will get a minimal surface S with this perturbed boundary. This minimal surface S will have the propertythat the tangent planes at the interior of β are not vertical, by the maximumprinciple with boundary.Applying translations along the x -axis and y -axis, we can use the trans-lates of S to show that Σ n is under S in a neighborhood of α , and then wehave uniform gradient estimates for points in α . Analogously, constructingsimilar barriers, we can prove that we have uniform gradient estimates in aneighborhood of α . Observe that besides the gradient estimates, the translates of the minimalsurface S form a barrier for points in a neighborhood of α ∪ α . We have that Σ n is a monotone increasing sequence of minimal graphswith uniform gradient estimates in α ∪ α , and it is a bounded graph forpoints in a neighborhood U of the origin (because of the barrier given by theannulus A ). Therefore, there exists a subsequence of Σ n that converges to a ERIODIC MINIMAL SURFACES IN SEMIDIRECT PRODUCTS (cid:101)
Σ with α ∪ α in its boundary. As we already know that Σ n converges to the minimal surface Σ , we conclude that in fact Σ = (cid:101) Σ , and thenΣ is a minimal surface with α ∪ α ∪ α ∪ α in its boundary. Notice that wecan assume that Σ has P ∞ as its boundary, with Σ being of class C up to P ∞ \ { ( a, , , (0 , a, } and continuous up to P ∞ . Now considering the rotation by angle π around α of Σ , we obtain thesurface illustrated in Figure 3.Figure 3: Rotation by angle π around α of Σ . Continuing to rotate by angle π around the y -axis, the resulting surfacewill be a minimal surface with four vertical lines as its boundary: { ( a, , t ) : t ∈ R } , { (0 , a, t ) : t ∈ R } , { ( − a, , t ) : t ∈ R } , { (0 , − a, t ) : t ∈ R } . Now we can use the rotations by angle π around the vertical lines to geta complete minimal surface that is analogous to the doubly periodic minimalScherk surface in R . It is invariant by two translations that commute and it isa four punctured sphere in the quotient of R (cid:111) A R by the group of isometriesgenerated by the two translations. Theorem 2.
In any semidirect product R (cid:111) A R , where A = (cid:18) bc (cid:19) , thereexists a periodic minimal surface similar to the doubly periodic Scherk minimalsurface in R . Throughout this section, we consider the semidirect product R (cid:111) A R withthe canonical left invariant metric (cid:104) , (cid:105) , where A = (cid:18) bc (cid:19) . In this space,we construct a complete minimal surface similar to the singly periodic Scherkminimal surface in R .2 A. MENEZES
Fix c > < (cid:15) < a sufficiently smalls so that a + 2 (cid:15) < (cid:90) c (cid:113) a ( z ) + a ( z ) dz. For each c > , consider the polygon P c in R (cid:111) A R with the six sidesdefined below. α c = { ( t, ,
0) : 0 ≤ t ≤ c } α c = { ( c, t,
0) : 0 ≤ t ≤ a } α c = { ( t, a,
0) : 0 ≤ t ≤ c } α c = { (0 , a, t ) : 0 ≤ t ≤ c } α c = { (0 , t, c ) : 0 ≤ t ≤ a } α c = { (0 , , t ) : 0 ≤ t ≤ c } , and for each δ > δ < a/ , consider the polygon P δc with the followingsix sides. α δ,c = (cid:8) ( t, δc t,
0) : 0 ≤ t ≤ c (cid:9) α δ,c = { ( c, t,
0) : δ ≤ t ≤ a − δ } α δ,c = (cid:8) ( t, ac − δtc ,
0) : 0 ≤ t ≤ c (cid:9) ,α c , α c , α c , as illustrated in Figure 4.Figure 4: Polygons P c and P δc . Denote by Ω( δ, c ) the region in R (cid:111) A { } bounded by α δ,c , α δ,c , α δ,c andthe segment { (0 , t,
0) : 0 ≤ t ≤ a } . For each c and δ, we have compact minimalsurfaces Σ c and Σ δc with boundary P c and P δc , respectively, which are solutionsto the Plateau problem. By Theorem 1, we know that Σ c and Σ δc are stable ERIODIC MINIMAL SURFACES IN SEMIDIRECT PRODUCTS , c ) , Ω( δ, c ) , respectively. We willshow that Σ c is the unique compact minimal surface with boundary P c . Fix c. For each 0 < δ < a/ , P δc is a polygon transverse to the Killing field ∂ x and each integral curve of ∂ x intersects P δc in at most one point. Thus wecan prove, as we did in Proposition 3, that Σ δc is the unique compact minimalsurface with boundary P δc . Denote by u δc , v c the functions defined in the interior of Ω( δ, c ) , Ω(0 , c ) , whose Π-graphs are Σ δc , Σ c , respectively. Then, as ∂ x is a Killing field and each P δc is transversal to ∂ x , we can use the flow of ∂ x and the maximum principleto prove that for δ (cid:48) < δ we have 0 ≤ u δc ≤ u δ (cid:48) c ≤ v c in intΩ( δ, c ) , hence v c isa barrier for our sequence u δc . Because of the monotonicity and the barrier,the family u δc converges to a function u c defined in intΩ(0 , c ) whose graph isa compact minimal surface with boundary P c , and we still have u c ≤ v c onΩ(0 , c ) . Now we will find another compact minimal surface with boundary P c , whose interior is the graph of a function w c defined in intΩ(0 , c ) such that v c ≤ w c and we will show that u c = w c . In order to do that, for each 0 < δ
0) : 0 ≤ t ≤ c } α c = { ( c, t, 0) : 0 ≤ t ≤ a } (cid:101) α δ,c = { ( t, ( a + δ ) c − δtc , 0) : 0 ≤ t ≤ c } (cid:101) α δ,c = { (0 , a + δ, t ) : 0 ≤ t ≤ c } (cid:101) α δ,c = { (0 , t, c ) : − δ ≤ t ≤ a + δ } (cid:101) α δ,c = { (0 , − δ, t ) : 0 ≤ t ≤ c } . Figure 5: Polygons P c and (cid:101) P δc . A. MENEZES Denote by (cid:101) Ω( δ, c ) the region in R (cid:111) A { } bounded by (cid:101) α δ,c , α c , (cid:101) α δ,c andthe segment { (0 , t, 0) : − δ ≤ t ≤ a + δ } . For each δ, we have a compact minimaldisk (cid:101) Σ δc with boundary (cid:101) P δc and (cid:101) Σ δc is a smooth Π-graph over the interior of (cid:101) Ω( δ, c ) . As (cid:101) P δc is transversal to the Killing field ∂ x , we can prove that (cid:101) Σ δc isthe unique compact minimal surface with boundary (cid:101) P δc . Denote by w δc the function defined in int (cid:101) Ω( δ, c ) whose graph is (cid:101) Σ δc . Usingthe flow of ∂ x and the maximum principle, we can prove that for δ (cid:48) < δ wehave w δ (cid:48) c ≤ w δc in int (cid:101) Ω( δ (cid:48) , c ) and for all δ, v c ≤ w δc in intΩ(0 , c ) . Because ofthe monotonicity and the barrier, the family w δc converges to a function w c defined in int (cid:101) Ω(0 , c ) = intΩ(0 , c ) whose graph is a compact minimal surfacewith boundary P c , and we still have v c ≤ w c in intΩ(0 , c ) . Let us call Σ , Σ the graphs of u c , w c , respectively. We will now provethat Σ = Σ . Denote by ν i the conormal to Σ i along P c , i = 1 , . (See Figure6). Figure 6: Σ and Σ . Suppose that u c (cid:54) = w c , then in fact we have u c < w c in intΩ(0 , c ) . As ∂ x istangent to α c and α c , then (cid:104) ν i , ∂ x (cid:105) = 0 , i = 1 , , in α c and α c . In the othersides of P c we have (cid:104) ν , ∂ x (cid:105) < (cid:104) ν , ∂ x (cid:105) . Therefore, (cid:90) P c (cid:104) ν , ∂ x (cid:105) < (cid:90) P c (cid:104) ν , ∂ x (cid:105) . But, using the Flux Formula for Σ and Σ with respect to the Killing field ∂ x , we have (cid:90) P c (cid:104) ν , ∂ x (cid:105) = 0 = (cid:90) P c (cid:104) ν , ∂ x (cid:105) . Then, u c = w c and therefore, Σ c = Σ = Σ . In particular, Σ c is the uniquecompact minimal surface with boundary P c . Denote by Ω( ∞ ) the infinite strip { ( x, y, 0) : x ≥ , ≤ y ≤ a } , andby R the region { ( x, y, z ) : x ≥ , ≤ y ≤ a, z ≥ } . Moreover, denote α = { ( x, , 0) : x > } , α = { ( x, a, 0) : x > } , α = { (0 , a, z ) : z > } and α = { (0 , , z ) : z > } , hence P ∞ = α ∪ α ∪ α ∪ α ∪ { (0 , , , (0 , a, } . ERIODIC MINIMAL SURFACES IN SEMIDIRECT PRODUCTS n ∈ N , let Σ n be the unique compact minimal surface with bound-ary P n . We are interested in proving the existence of a subsequence of Σ n thatconverges to a complete minimal surface with boundary P ∞ . Using the ex-istence of a minimal annulus, guaranteed by the Douglas criterion, we willshow that there exist points p n ∈ Σ n , Π( p n ) = q ∈ int Ω( ∞ ) for all n, whichconverge to a point p ∈ R (cid:111) A R , and then we will use Proposition 4.Consider the parallelepiped with faces A, B, C, D, E and F, defined below. A = { ( u, − (cid:15), v ) : (cid:15) ≤ u ≤ d ; 0 ≤ v ≤ c } B = { ( u, a + (cid:15), v ) : (cid:15) ≤ u ≤ d ; 0 ≤ v ≤ c } C = { ( u, v, 0) : (cid:15) ≤ u ≤ d ; − (cid:15) ≤ v ≤ a + (cid:15) } D = { ( u, v, c ) : (cid:15) ≤ u ≤ d ; − (cid:15) ≤ v ≤ a + (cid:15) } E = { ( (cid:15), u, v ) : − (cid:15) ≤ u ≤ a + (cid:15) ; 0 ≤ v ≤ c } F = { ( d, u, v ) : − (cid:15) ≤ u ≤ a + (cid:15) ; 0 ≤ v ≤ c } , where d > (cid:15) is a constant that we will choose later.As we did in the last section, we can calculate the area of each one of thesefaces and we obtain:area A = area B = ( d − (cid:15) ) (cid:90) c (cid:113) a ( z ) + a ( z )d z, area C = area D = ( d − (cid:15) )( a + 2 (cid:15) ) , area E = area F = ( a + 2 (cid:15) ) (cid:90) c (cid:113) a ( z ) + a ( z )d z. Hence, area C + area D + area E + area F < area A + area B se, e somente se,( d − (cid:15) )( a + 2 (cid:15) ) + ( a + 2 (cid:15) ) (cid:90) c (cid:113) a + a d z < ( d − (cid:15) ) (cid:90) c (cid:113) a + a d z se, e somente se,( d − (cid:15) ) (cid:20) ( a + 2 (cid:15) ) − (cid:90) c (cid:113) a + a d z (cid:21) < − ( a + 2 (cid:15) ) (cid:90) c (cid:113) a + a d z se, e somente se, d > (cid:15) − ( a + 2 (cid:15) ) (cid:90) c (cid:113) a ( z ) + a ( z )d z ( a + 2 (cid:15) ) − (cid:90) c (cid:113) a ( z ) + a ( z )d z . A. MENEZES As we chose a + 2 (cid:15) < (cid:90) c (cid:113) a ( z ) + a ( z )d z, we can choose d > (cid:15) so thatthe Douglas criterion is satisfied [1]. Thus, there exists a minimal annulus A with boundary ∂A ∪ ∂B such that its projection Π( A ) contains points ofintΩ( ∞ ) . (See Figure 7). Figure 7: Annulus A .We know that, for each c < (cid:15), Σ c ∩ A = ∅ . When c increases P c does notintersect ∂ A , then, using the maximum principle, Σ c ∩ A = ∅ for all c, andΣ c is under the annulus A . Thus, there exists a point q ∈ intΩ( ∞ ) such that p n = Σ n ∩ Π − ( q ) has a subsequence that converges to a point p ∈ Π − ( q ) . Observe that applying the flow of the Killing field ∂ x to the annulus A we canconclude that, in the region { x ≥ d } , the surfaces Σ n are bounded above by,for example, the plane { z = c } . In order to understand the convergence of the surfaces Σ n we need to provesome properties of these surfaces. Claim 2. The surfaces Σ n are transversal to the Killing field ∂ x in the interior.Proof. Fix n. Suppose that at some point p ∈ intΣ n the tangent plane T p Σ n contains the vector ∂ x . As the planes that contain the direction ∂ x are minimalsurfaces, we have that Σ n and T p Σ n are minimal surfaces tangent at p, andthen the intersection between them is formed by 2 k curves, k ≥ , passingthrough p making equal angles at p. By the shape of P n (the boundary of Σ n ),we know that T p Σ n intersects P n either in only two points or in one point anda segment of straight line ( α n or α n ). Therefore, we will have necessarily aclosed curve contained in the intersection. As Σ n is simply connected this curvebounds a disk in Σ n , but as the parallel planes to T p Σ n are minimal surfaces,we can use the maximum principle to prove that this disk is contained in theplane T p Σ n and then they coincide, which is impossible. Thus, the vector ∂ x is transversal to Σ at points p ∈ intΣ n . ERIODIC MINIMAL SURFACES IN SEMIDIRECT PRODUCTS n are also transver-sal to ∂ x at the points in α and α , by the maximum principle with boundary.Thus rotation by angle π around α (respectively α ) gives a minimal surfacewhich is also transversal to the Killing field ∂ x in the interior, extends thesurface Σ n and has α n (respectively α n ) in the interior. Therefore, we haveuniform curvature estimates for Σ n up to α ∪ α . Hence, for every compact contained in { z > } ∩ R , there exists a subse-quence of Σ n that converges to a minimal surface. Taking an exhaustion bycompact sets and using a diagonal process, we conclude that there exists asubsequence of Σ n that converges to a minimal surface Σ that has α ∪ α inits boundary. From now on we will use the notation Σ n for this subsequence.It remains to prove that in fact Σ is a minimal surface with boundary P ∞ . In order to do that, we will use the fact that each Σ n is a vertical graph inthe interior. Let us denote by u n the function defined in intΩ( n ) such thatΣ n = Graph( u n ) , where Ω( n ) = { ( x, y, 0) : 0 ≤ x ≤ n ; 0 ≤ y ≤ a } . Claim 3. u n − < u n in intΩ( n − . Proof. Recall that each Σ n is the limit of a sequence of minimal graphs (cid:101) Σ δn =Graph( w δn ) whose boundary is transversal to the Killing field ∂x. Using theflow of the Killing field ∂ x , we can prove that each (cid:101) Σ δn is above Σ n − , and thenthe limit surface Σ n has to be above Σ n − . In fact, Σ n is strictly above Σ n − in the interior, because as Σ n and Σ n − are minimal surfaces, if they intersectat an interior point, there will be points of Σ n under Σ n − , and we alreadyknow that, by the property of (cid:101) Σ δn , this is not possible. Claim 4. There are uniform gradient estimates for { u n } for points in α ∪ α . Proof. We will use the same idea as in Claim 1. For y > a and δ > β = { ( x, y , c ) : d ≤ x ≤ d + δ } ,β = { ( t, y , c d t ) : 0 ≤ t ≤ d } , β = { ( t + δ, y , c d t ) : 0 ≤ t ≤ d } and β = { ( x, y , 0) : 0 ≤ x ≤ δ } . This is a minimal surface transversal to the Killingfield ∂ y , hence any small perturbation of its boundary gives a minimal surfacewith that perturbed boundary. Thus, if we consider a small perturbation ofthe boundary of this vertical strip by just perturbing a little bit β by a curvecontained in { y ≤ y } joining the points ( d, y , c ) and ( d + δ, y , c ) , we willget a minimal surface S with this perturbed boundary. This minimal surface S will have the property that the tangent planes at the interior points of β are not vertical, by the maximum principle with boundary.Applying translations along the x -axis and y -axis, we can use the trans-lates of S to show that Σ n is under S in a neighborhood of α , and then wehave uniform gradient estimates for points in α . Analogously, constructingsimilar barriers, we can prove that we have uniform gradient estimates in aneighborhood of α . Observe that besides the gradient estimates, the translates of the minimalsurface S form a barrier for points in a neighborhood of α ∪ α . A. MENEZES We have that Σ n is a monotone increasing sequence of minimal graphswith uniform gradient estimates in α ∪ α , and it is a bounded graph forpoints in { x ≥ d } (because of the barrier given by the annulus A ). Therefore,there exists a subsequence of Σ n that converges to a minimal surface (cid:101) Σ with α ∪ α in its boundary. As we already know that Σ n converges to the minimalsurface Σ , we conclude that in fact Σ = (cid:101) Σ , and then Σ is a minimal surfacewith α ∪ α ∪ α ∪ α in its boundary. Notice that we can assume that Σ has P ∞ as its boundary, with Σ being of class C up to P ∞ \{ (0 , , , (0 , a, } andcontinuous up to P ∞ . The expected “singly periodic Scherk minimal surface”is obtained by rotating recursively Σ by an angle π about the vertical andhorizontal geodesics in its boundary. Theorem 3. In any semidirect product R (cid:111) A R , where A = (cid:18) bc (cid:19) , thereexists a periodic minimal surface similar to the singly periodic Scherk minimalsurface in R . References [1] J. Jost. Conformal mapping and the Plateau-Douglas problem in Rieman-nian manifolds . J. Reine Angew. Math (1985), 37–54.[2] L. Mazet, M. Magdalena Rodr´ıguez and H. Rosenberg. Periodic constantmean curvature surfaces in H × R . Preprint (2011), arXiv:1106.5900.[3] W. H. Meeks III, P. Mira, J. P´erez and A. Ros. Constant mean curvaturespheres in homogeneous three-manifolds. Work in progress.[4] W. H. Meeks and J. P´erez. Constant mean curvature surfaces in metric Liegroups. Geometric Analysis: Partial Differential Equations and Surfaces,Contemporary Mathematics (AMS) (2012), 25–110.[5] H. Rosenberg. Minimal surfaces in M × R . Illinois J. Math. (2002),1177–1195.[6] H. Rosenberg, R. Souam and E. Toubiana. General curvature estimates forstable H -surfaces in 3-manifolds and aplications. J. Differential Geom. (2010), 623–648. Instituto Nacional de Matem´atica Pura e Aplicada (IMPA)Estrada Dona Castorina 110, 22460-320, Rio de Janeiro-RJ,Brazil