Classical Schwarz Reflection Principle for Jenkins-Serrin Type Minimal Surfaces

We give a proof of the classical Schwarz reflection principle for Jenkins-Serrin type minimal surfaces in the homogeneous three manifolds $E(\kappa, \tau)$ for $\kappa \leqslant 0$ and $\tau \geqslant 0$. In our previous paper we proved a reflection principle in Riemannian manifolds. The statements and techniques in the two papers are distinct.


Introduction
In this paper we focus the classical Schwarz reflection principle across a geodesic line in the boundary of a minimal surface in R 3 and more generally in three dimensional homogeneous spaces E(κ, τ ) for κ < 0 and τ 0.
The Schwarz reflection principle was shown in some special cases.One kind of examples arise for the solutions of the classical Plateau problem in R 3 containing a segment of a straight line in the boundary, see Lawson [8, Chapter II, Section 4, Proposition 10].Another kind occur for vertical graphs in R 3 and H 2 × R containing an arc of a horizontal geodesic, see [20,Lemma 3.6].
On the other hand, there is no proof of the reflection principle for general minimal surfaces in R 3 containing a straight line in its boundary.
The goal of this paper is to provide a proof of the reflection principle about vertical geodesic lines for Jenkins-Serrin type minimal surfaces in R 3 and other three dimensional homogeneous manifolds such as, for example, H 2 × R, PSL 2 (R, τ ) and S 2 × R, see Theorem 4.1.The proof also holds for horizontal geodesic lines.
We observe that this classical Schwarz reflection principle was used by many authors, including the present authors, in R 3 and H 2 × R.
We recall that the authors proved another reflection principle for minimal surfaces in general three dimensional Riemannian manifold with quite different statement and techniques, see [21].
We are grateful to the referee of our paper whose remarks greatly improved this work.

A brief description of the three dimensional
homogeneous manifolds E(κ, τ ) For any r > 0 we denote by D(r) ⊂ R 2 the open disc of R 2 with center at the origin and with radius r (for the Euclidean metric).
We denote by M(κ) the complete, connected and simply connected Riemannian surface with constant curvature κ.Notice that for κ < 0 a model of M(κ) is given by the disc D( 2 √ −κ ) equipped with the metric ν 2 κ (dx 2 + dy 2 ).We recall that E(κ, τ ) is a fibration over M(κ), and the projection Π : E(κ, τ ) −→ M(κ) is a Riemannian submersion, see for example [2].Moreover the unit vertical field ∂ ∂t is a Killing field generating a one-parameter group of isometries given by the vertical translations.
We have seen in [21,] that the horizontal geodesics and the vertical geodesics of E(κ, τ ) admit a reflection.That is, for any such a geodesic L, there exists a non trivial isometry Let Ω be any domain of M(κ) and let u : Ω −→ R be a C 2 -function.
We say that the set Σ : Note that the Killing field ∂ ∂t is transverse to Σ. Thus, by the well-known criterium of stability, if Σ is a minimal surface then Σ is stable.Consider some arbitrary local coordinates (x 1 , x 2 , x 3 ) of E(κ, τ ).Let u be a C 2 function defined on a domain Ω contained in the x 1 , x 2 plane of coordinates.Let S ⊂ E(κ, τ ) be the graph of u.Then S is a minimal surface if u satisfies an elliptic PDE (called minimal surface equation) see [21,Equation (13)].Furthermore, if u has bounded gradient then the PDE is uniformly elliptic.

Jenkins-Serrin type minimal surfaces
The original Jenkins-Serrin's theorem was conceived in R  [13,Section 3.6].As a matter of fact the same proof also works in the homogeneous spaces E(κ, τ ) for any κ < 0 and τ 0.
We state briefly below the Jenkin-Serrin type theorem in the homogeneous spaces E(κ, τ ) for κ < 0 and τ 0 (same statement holds in R 3 and in M 2 × R).
Let Ω be a bounded convex domain in M 2 (κ), thus for any point p ∈ Γ := ∂Ω there is a complete geodesic line We assume that the C 0 Jordan curve Γ ⊂ M 2 (κ) is constituted of two families of open geodesic arcs A 1 , . . ., A a , B 1 , . . ., B b and a family of open arcs C 1 , . . ., C c with their endpoints.We assume also that no two A i and no two B j have a common endpoint.
On each open arc C k we assign a continuous boundary data g k .Let P ⊂ Ω be any polygon whose vertices are chosen among the endpoints of the open geodesic arcs A i , B j , we call P an admissible polygon.We set With the above notations the Jenkins-Serrin's theorem asserts the following: If the family {C k } is not empty then there exists a function u : Ω −→ R whose graph is a minimal surface in E(κ, τ ) and such that u 2α(P ) < γ(P ), 2β(P ) < γ(P ) for any admissible polygon P .In this case the function u is unique.
If the family {C k } is empty such a function u exists if and only if α(Γ) = β(Γ) and condition (2) holds for any admissible polygon P = Γ.In this case the function u is unique up to an additive constant.
We denote by Σ ⊂ E(κ, τ ) the graph of u over Ω and we call such a surface a Jenkins-Serrin type minimal surface.
Remark 3.1.We observe that when the family {C k } is empty, the boundary of Σ is the union of vertical geodesic line {q} × R for any common endpoint q between geodesic arcs A i and B j .
Suppose that the family {C k } is not empty and let x 0 be a common vertex between A i and C k , if any.If g k has a finite limit at x 0 , say α, then the half vertical line {x 0 } × [α, +∞[ lies in the boundary of Σ.Now if x 0 is a common vertex between B j and C k and if g k has a finite limit at x 0 , say β, then the half vertical line {x 0 }× ] − ∞, β] lies in the boundary of Σ.At last, if x 0 is a common vertex between C i and C k and if g i and g k have different finite limits at x 0 , say α < β, then the vertical segment {x 0 } × [α, β] lies in the boundary of Σ.

Main theorem
For any vertical geodesic line L of E(κ, τ ), we denote by I L the reflection about the line L.Then, we can extend minimally Σ by reflection about L.More precisely, S := Σ ∪ γ ∪ I L (Σ) is a smooth minimal surface invariant by the reflection about Γ, containing int(γ) in its interior.
Furthermore the same statement and proof hold for Observe that the possible cases for γ are the following: the whole line L, a half line of L or a closed geodesic arc of L.
Observe also that, since we are under the assumptions of Remark 3.1, if x 0 is an endpoint of some arc C i , then g i has a finite limit at x 0 .Remark 4.2.We use the same notations as in Theorem 4.1.Suppose that the boundary of Σ contains an open arc δ (graph over an arc C k ) of a horizontal geodesic line Υ of PSL 2 (R, τ ).
We denote by I Υ the reflection in PSL 2 (R, τ ) about Υ.
We can prove as in [20,Lemma 3.6] (in H 2 × R) that we can extend Σ by reflection about Υ: Σ ∪ δ ∪ I Υ (Σ) is a connected smooth minimal surface containing δ in its interior.The same observation holds also in Heisenberg space and S 2 × R.
On the other hand, we can verify that the proof of Theorem 4.1 also works for reflection about horizontal geodesic lines.
We assume that the family C k is not empty.The other situation can be handled in a similar way.
Recall that, by assumption, if x 0 is an endpoint of some arc C i (if any), then g i has a finite limit at x 0 .
We suppose that all functions g k admit also a finite limit at the endpoints of C k different of x 0 (if any).It is possible to carry out a proof without this assumption but the details are cumbersome, as we can see in the following.
Suppose for instance that x 1 ( = x 0 ) is an endpoint of some arc C k , that g k has no limit at x 1 and that g k is bounded near Then we consider the new function g k,n on C k setting g k,n (x) = α on the segment [x 1 , p n ] of C k and g k,n = g k outside this segment.Now the continuous function g k,n has a limit at x 1 .Observe that for any x ∈ C k we have g k,n (x) = g k (x) for any n large enough.
If g k is not bounded near x 1 , we first truncate, for any n > 0, the function g k above by n and below by -n.We obtain a new continuous and bounded function h k,n on C k .Then we proceed as above.
For any integer n we consider the Jordan curve Γ n obtained by the union of the geodesic arcs A i at height n, the geodesic arcs B j at height −n, the graphs of functions g k over the open arcs C k (or g k,n if g k has no finite limit at some endpoint of C k ), and the vertical segments necessary to form a Jordan curve.Thus Γ is the projection of Γ n on H 2 .Let Σ n ⊂ PSL 2 (R, τ ) be the embedded area minimizing disc with boundary Γ n given by Proposition 6.1 in the Appendix.We have We set γ n := Σ n ∩ L, where L := {x 0 } × R, thus γ n ⊂ γ for any n.Due to the fact that Σ n is area minimizing we can apply the reflection principle about the vertical line L, this is proven in detail in [21,Proposition 3.4].Thus, S n := Σ n ∪ I L (Σ n ) is an embedded minimal surface containing int(γ n ) in its interior.By construction S n is invariant under the reflection I L and is orientable.
Let u n : Ω −→ R be the function whose the graph is Σn .Thus u n extends continuously by n on the edges int(A i ), by −n on the edges int(B j ) and by g k (or g k,n ) over the open arcs C k .Using the lemmas derived in [25], following the original proof of [7, Theorem 2], it can be proved that, up to considering a subsequence, the sequence of functions (u n ) converges to a function u : Ω −→ R in the C 2 -topology, uniformly over any compact subset of Ω.
Let d n be the intrinsic distance on S n .For any p ∈ S n and any r > 0 we denote by B n (p, r) ⊂ S n the open geodesic disc of S n centered at p with radius r.By construction, for any p ∈ int(γ) there exist n p ∈ N and a real number c p > 0 such that for any integer n n p we have We assert that the Gaussian curvature K n of the surfaces S n is uniformly bounded in the neighborhood of each point of int(γ), independently of n.Proposition 4.3.For any p ∈ int(γ) there exist R p , K p > 0, and there exists n p ∈ N satisfying p ∈ int(γ np ) ⊂ S np and d np (p, ∂S np ) > 2R p , such that for any integer n n p we have p ∈ int(γ n ) ⊂ S n and We postpone the proof of Proposition 4.3 until Section 5. Assuming Proposition 4.3 we will prove that for any p ∈ int(γ) there exists an embedded minimal disc D(p), containing p in its interior, such that D(p) ⊂ Σ ∪ γ ∪ I L (Σ).This will prove that Σ ∪ γ ∪ I L (Σ) is a minimal surface, that is smooth along int(γ).
Let p ∈ int(γ), we deduce from Proposition 4.3 that there exist real numbers R p , K p > 0 and n p ∈ N such that for any integer n n p and for any point Therefore it can be proved as in [18], using [18, Proposition 2.3, Lemma 2.4] and the discussion that follows, that up to taking a subsequence, the geodesic discs B n (p, R p ) converge for the C 2 -topology to a minimal disc D(p) ⊂ R 3 containing p in its interior.We recall that each geodesic disc B n (p, R p ) is embedded, contains an open subarc γ(p) of γ (which does not depend on n) passing through p, and B n (p, R p ) is invariant under the reflection I L .Thereby the minimal disc D(p) also is embedded, contains the subarc γ(p) and inherits the same symmetry.
We set Then we have D(p) ⊂ S. We conclude henceforth that S is a smooth minimal surface invariant under the reflection I L , this accomplishes the proof of the theorem.
Remark 4.4.Theorem 4.1 holds also in case where x 0 is and endpoint of some arc C i and g i has an infinite limit at x 0 .
Indeed, assume that lim x→x 0 g i (x) = +∞.We denote by g i,n the new function on C i obtained by truncating the function g i above by n.Then, in the proof of Theorem 4.1, we consider the embedded area minimizing disc Σ n constructed with the function g i,n on C i (instead of g i ).Then we can proceed the proof in the same way.
Remark 4.5.We don't know if the Jenkins-Serrin type theorem was established in the Heisenberg spaces Nil 3 (τ ) = E(0, τ ) for τ > 0. Assuming the Jenkins-Serrin type theorem, the proof of Theorem 4.1 works to establish the same reflection principle for vertical geodesic lines in Nil 3 (τ ).

Proof of Proposition 4.3
We argue by absurd.Suppose by contradiction that there exists p ∈ int(γ) such that for any k ∈ N * there exist an integer n k > k and There exist c > 0 and k 0 ∈ N * such that for any integer k k 0 we have p ∈ int(γ n k ) and Moreover there exists an integer k 1 > k 0 such that for any integer k k 1 we have From now on, we are going to use classical blow-up techniques.Define the continuous function Clearly f k ≡ 0 on ∂B n k (p, c) and We fix a point p k ∈ B n k (p, c) where the function f k attains its maximum value, hence We deduce therefore (4) Notice that D k is embedded.
For further purpose we emphasize that D k is an orientable minimal surface of PSL 2 (R, τ ).

We denote by K
We summarize some facts derived before: Lemma 5.3.
• each D k is an embedded and orientable minimal surface of (D(2λ k ) × R, , • the metrics ds 2 k converge to g euc for the C 2 -topology, uniformly on any compact subset of R 3 , see (6).
Therefore it can be proved as in [18] (using [18, Proposition 2.3, Lemma 2.4] and the discussion that follows), that up to considering a subsequence, the D k converge for the C 2 -topology to a complete, connected and orientable minimal surface S of R 3 .
Remark 5.4.From the construction described in [18], the surface S has the following properties.
There exist r, r 0 > 0 such that for any q ∈ S, a piece G(q) of S, containing the geodesic disc with center q and radius r 0 , is a graph over the open disc D(q, r) of T q S with center q and radius r (for the Euclidean metric of R 3 ).Furthermore: a graph over D(q, r) and the surfaces G k (q) converge for the C 2 -topology to G(q), • for any y ∈ G(q) there exists k y ∈ N such that for any k k y we can choose the piece By construction we have 0 3 ∈ S and, denoting K the Gaussian curvature of S in (R 3 , g euc ), we deduce from (9) (10) | K(0 3 )|= 1.
For any integer k k 1 we set L k := H k (L).Thus, L k is a vertical straight line of R 3 .Definition 5.5.Let δ k be the distance in D(2λ k ) × R induced by the metric ds 2 k .We say that the sequence of vertical lines ( L k ) in R 3 disappears to infinity if δ k (0 3 , L k ) → +∞ when k → +∞ There are two possibilities: the sequence ( L k ) disappears or not to infinity.We are going to show that either case cannot occur, we will find therefore a contradiction.First case: ( L k ) disappears to infinity.
Observe that, by construction, the geodesic discs B n k (p, c) are invariant under the reflection I L and f k (q) = f k (I L (q)) for any q ∈ B n k (p, c).Since p k = 0 3 by assumption, we can assume that 0 Let q ∈ S, and consider a minimizing geodesic arc δ ⊂ S joining 0 3 to q.It follows from Remark 5.4 that there exist a finite number of points q 1 = 0 3 , . . ., q n = q belonging to δ, and there exists k q ∈ N such that: • for any integer k k q the subset ∪ j G k (q j ) ⊂ D k is connected and converges for the C 2 -topology to the subset ∪ j G(q j ) ⊂ S, • for any integer k k q we have ∪ j G k (q j ) ∩ L k = ∅.
Thus for any integer k k q we obtain that . Therefore it can be proved as in the discussion following Lemma 2.4 in [18] that S is a connected, complete, orientable and stable minimal surface of R 3 .Thanks to results of do Carmo and Peng [3], Fischer-Colbrie and Schoen [4] and Pogorelov [16], S is a plane.But this gives a contradiction with the curvature relation (10).
Second case: ( L k ) does not disappear to infinity.
We will prove that the Gauss map of S omits infinitely many points, hence S would be a plane (see [5,Corollary 1.3] or [24]), contradicting the curvature relation (10).
Let α ∈ (0, π] be the interior angle of Γ at vertex x 0 , (α exists since Γ is the boundary of a convex domain).Observe that the case where α = π is under consideration.
Since Ω is convex, there exists a geodesic line When τ = 0 notice that Π, is a vertical totally geodesic plane in H 2 × R. We recall that there are no totally geodesic surfaces in E(−1, τ ) if τ = 0, see [23,Theorem 1].
Under our assumption, up to considering a subsequence, we can assume that the sequence ( L k ) converges to a vertical straight line L ⊂ R 3 and that H k (Π) converges to a vertical plane Π ⊂ R 3 containing L. Let us denote by Π + and Π − the two open halfspaces of R 3 bounded by Π. Lemma 5.6.We have Proof.Otherwise assume there exists a point q ∈ S ∩ Π such that q ∈ L. From the structure of the intersection of two minimal surfaces tangent at a point, see [1,Theorem 7.3] or [22,Lemma,p.380], we may suppose that Π is transverse to S at q. Thus there is an open piece F (q) of S containing q which is transverse to the plane Π.Hence, for any integer k large enough, a piece For any integer k k 1 we denote by N k a smooth unit normal vector field on D k with respect to the metric ds 2 k , see (6).Let N k 3 be the vertical component of N k , this means that N k − N k 3 ∂ ∂t and ∂ ∂t are orthogonal vector fields along D k .
Since ( D k ) converges to S for the C 2 -topology, we can define a unit normal field N on S as the limit of the fields N k .Lemma 5.9.We have N 3 = 0 on S + ∪ S − .Furthermore S + and S − are vertical graphs.
Proof.Indeed, we know that D + k is a vertical graph.So we can assume that N k 3 > 0 along D + k for any k k 1 .By considering the limit of the fields N k we get that N 3 0 on S + .
Let q ∈ S + be a point such that N 3 (q) = 0, if any.Recall that the Gauss map of a non planar minimal surface of R 3 is an open map.Therefore, in any neighborhood of q in S + it would exist points y ∈ S + such that N 3 (y) < 0, which leads to a contradiction.
Thus we have N 3 = 0 on S + .We prove in the same way that N 3 = 0 on S − too.
Assume by contradiction that S + is not a vertical graph.Then there exist two points q, q ∈ S + lying to same vertical straight line.As the tangent planes of S + at q and q are not vertical, there exists a real number δ > 0 such that a neighborhood V q ⊂ S + of q and a neighborhood V q ⊂ S + of q are vertical graphs over an Euclidean disc of radius δ in the (u, v)-plane.
But, by construction, for k large enough a piece U q of D + k is C 2 -close of V q and a piece U q of D + k is C 2 -close of V q .Clearly this would imply that the vertical projections of U q and U q on the (u, v)-plane have non empty intersection.But this is not possible since D + k is a vertical graph.We conclude therefore that S + is a vertical graph.
We can prove in the same way that S − is a vertical graph.

End of the proof of the proposition
Let P ⊂ R 3 be any vertical plane verifying L ⊂ P and P = Π.We deduce from Lemmas 5.7 and 5.9 that ( S ∩ P ) \ L is a vertical graph.Therefore, the structure of the intersection of two minimal surfaces tangent at a point, see [1,Theorem 7.3] or [22,Lemma,p. 380], shows that there cannot be two distinct points of L where the tangent plane of S is P .reflection principle around γ to Σ n , see [21,Proposition 3.4].Then we proceed as in the proof of Proposition 4.3.
Observe that the same result holds also for S 2 × R.

Appendix
Let Ω ⊂ M(κ), κ 0, be a bounded convex domain bounded by a C 0 Jordan curve Γ := ∂Ω, and let f : Γ −→ R be a piecewise continuous function, allowing a finite number of discontinuities.
We denote by Γ ⊂ E(κ, τ ), τ 0, the graph of f .Namely, if f is continuous then Γ is a Jordan curve with a one-to-one projection on Γ.If f has discontinuity points then Γ is constituted of a finite number of simple arcs admitting a one-to-one vertical projection on some subarc of Γ, and a vertical segment over each point of Γ where f is not continuous.
We consider also a C 0 bounded convex domain Ω in S 2 .In this case Γ ⊂ S 2 × R. Proof.We perform the proof in E(κ, τ ), the proof in S 2 ×R is analougous.Assume first that f has no discontinuity points.Since E(κ, τ ) is homogeneous, we deduce from a result of Morrey [10] that there exists a minimizing area disc Σ bounded by Γ.Since Ω is a convex domain, we deduce from the maximum principle that int(Σ) ⊂ Ω × R (we use also the fact that if γ ⊂ M(κ) is a geodesic line, then γ ×R is a minimal surface of E(κ, τ )).
Furthermore as Γ has a one-to-one vertical projection on Γ, we infer that int(Σ) is a vertical graph over Ω, as in Rado's theorem [8,Theorem 16].
Thus Σ is an embedded area minimizing disc bounded by Γ and is a vertical graph over Ω.Clearly, Σ is the unique minimal graph bounded by Γ.The same affirmation holds in S 2 × R.
Assume now that f has a finite number of discontinuity points, let x 0 ∈ Γ be such a point.Giving an orientation to Γ, we have therefore lim Let (p n ) be a sequence on Γ such that • p n → x 0 and p n > x 0 , • for any n, the function f is continuous on the arc (x 0 , p n ] of Γ.Now we modify f on the closed arc [x 0 , p n ] in such a way that the new function f n is strictly monotonous on [x 0 , p n ] and satisfies for any x ∈ (x 0 , p n ).
We assume that we have modified in the same way f in a neighborhood of each point of discontinuity and we continue denoting f n the new function.Clearly, we have We denote by Γ n the graph of f n .
We know from the beginning of the proof, that for any n there exists an (unique) embedded area minimizing disc Σ n bounded by the Jordan curve Γ n , which is a vertical graph over Ω.Let u n : Ω −→ R be the function whose the graph is Σ n .
Since the sequence (f n ) is uniformly bounded on Γ, we deduce from the maximum principle that the sequence (u n ) is uniformly bounded on Ω too.In [12, Section 4.1] it is stated a compactness principle in the case where the ambient space is Heisenberg space, but it can be stated and proved in the same way in E(κ, τ ), κ 0, τ 0, and also in S 2 × R.
Therefore, up to considering a subsequence, we can assume that the sequence of restricted maps (u n|Ω ) converges on Ω, for the topology C 2 and uniformly on any compact subset of Ω, to a function u : Ω −→ R satisfying the minimal surface equation.Thus the graph of u is a minimal surface Σ which has a one-to-one projection on Ω.
s n the bounded subset of the cylinder Γ × R bounded by the following curves: • the graph of f n over the arc [x 0 , p n ] of Γ, • the graph of f over the arc (x 0 , p n ], • the vertical closed segment above x 0 , with endpoints (x 0 , limx→x0 Clearly we have Area(s n ) → 0. Furthermore, by the foregoing discussion there exists ε > 0 such that Area(S) < Area(Σ n ) − ε for any n large enough.
Let {x i } ⊂ Γ be the finite set of points where f is not continuous.For any point x i we denote by s n (x i ) the piece of Γ × R constructed as above, corresponding to x i .
Observe now that S n := S ∪ (∪ i s n (x i )) is a disk with boundary the Jordan curve Γ n .By construction we have Area(S n ) < Area(Σ n ) for any n large enough, contradicting the fact that Σ n is a minimizing area disc with boundary Γ n .
Thus Σ is an embedded minimizing area disc with boundary Γ such that Σ \ Γ is a vertical graph over Ω.
Thus Σ 0 = Σ, which concludes the proof.Proof.We denote by x, y, z the coordinates on R 3 .Up to an isometry of R 3 we can assume that D is the x-axis: D = {(x, 0, 0), x ∈ R}.
By assumption there exist real numbers a < b such that (x, 0, 0) ∈ M for any x ∈ [a, b].
We are going to prove that A = −∞ and B = +∞ to conclude that D ⊂ M.
We have B b. Assume by contradiction that B = +∞.Since M is a complete surface we have (B, 0, 0) ∈ M. Let P ⊂ R 3 be the plane containing D and the orthogonal direction of M at (B, 0, 0).
Since the surfaces M and P are transverse at (B, 0, 0), their intersection in a neighborhood of (B, 0, 0) is an analytic arc γ.Furthermore, up to choosing a smaller arc, we can assume that γ is the graph, in P , of an analytic function f over the interval [B − ε, B + ε] for ε > 0 small enough.Since f is an analytic function satisfying f (x) = 0 for any x ∈ [B − ε, B], we deduce that f (x) = 0 for any x ∈ [B − ε, B + ε].Therefore we have (x, 0, 0) ∈ M for any x ∈ [a, B + ε], contradicting the definition of B.
Thus we have B = +∞.We prove in the same way that A = −∞, concluding the proof.
)) and we denote by D k ⊂ B n k (p, c) ⊂ S n k the open geodesic disc with center p k and radius ρ k /2.

Proposition 6 . 3 .
Let M ⊂ R 3 be a complete minimal surface containing a segment of a straight line D. Then the whole line D belongs to M: D ⊂ M.