aa r X i v : . [ m a t h . DG ] F e b On Delaunay Ends in the DPW Method
Thomas RaujouanFebruary 15, 2019
Abstract
We consider constant mean curvature 1 surfaces in R arising via the DPWmethod from a holomorphic perturbation of the standard Delaunay potential onthe punctured disk. Kilian, Rossman and Schmitt have proven that such a sur-face is asymptotic to a Delaunay surface. We consider families of such potentialsparametrised by the necksize of the model Delaunay surface and prove the existenceof a uniform disk on which the surfaces are close to the model Delaunay surface andare embedded in the unduloid case. Introduction
Beside the sphere, the simplest non-zero constant mean curvature (CMC) surface is thecylinder, which belongs to a one-parameter family of surfaces generated by the revolutionof an elliptic function: the Delaunay surfaces, first described in [1]. Like the cylinder,Delaunay surfaces have two annular type ends, and Delaunay ends are the only possibleembedded annular ends for a non-zero CMC surface. Indeed, as proven in [11] by Korevaar,Kusner and Solomon, if
M ⊂ R is a proper, embedded, non-zero CMC surface of finitetopological type, then every annular end of M is asymptotic to a Delaunay surface andif M has exactly two ends which are of annular type, then M is a Delaunay surface.Thus, the status of Delaunay surfaces for non-zero CMC surfaces is very much alike thecatenoid position in the study of minimal surfaces (see the result of Schoen in [17]), andone has to understand the behaviour of Delaunay ends in order to construct examples ofnon-compact CMC surfaces with annular ends, as Kapouleas did in 1990 [6].For an immersion, having a constant mean curvature and having a harmonic Gaussmap are equivalent. This is why the Weierstrass type representation of Dorfmeister, Peditand Wu [2] has been used since the publication of their article to construct CMC surfaces.The DPW method can construct any conformal non-zero CMC immersion in R , H or S n -noids and bubbletons, have been made by Dorfmeister, Wu,Killian, Kobayashi, McIntosh, Rossmann, Schmitt and Sterling [3, 16, 8, 9, 10, 15]. Theseconstructions often rely on a holomorphic perturbation of the holomorphic potential givingrise to a Delaunay surface via the DPW method, and Kilian, Rossmann and Schmitt [7]have proven that such perturbations always induce asymptotically a Delaunay end.More precisely, any Delaunay embedding can be obtained with a holomorphic potentialof the form ξ D = Az − dz where A = (cid:18) rλ − + srλ + s (cid:19) . The main result of [7] states that any immersion obtained from a perturbed potentialof the form ξ = ξ D + O ( z ) is asymptotic to an embedded half-Delaunay surface ina neighbourhood of z = 0 , provided that the monodromy problem is solved. In thispaper, we allow the perturbed potential to move in the family of Delaunay potentials byintroducing a real parameter t , proportional to the weight of the model Delaunay surface,and consider ξ t = ξ D t + O t ( z ) where ξ D t is a Delaunay potential of weight πt . Themain theorem of [7] states that for every t > , there exists a small neighbourhood of theorigin on which the surface produced by the potential ξ t is embedded and asymptotic toa half Delaunay surface. Unfortunately, without further hypotheses, this neighbourhoodvanishes into a single point as t tends to zero. Adding a few assumptions, we prove herethat there exists a uniform neighbourhood of the origin upon which the surfaces inducedby the family ξ t are all embedded and asymptotic to a half Delaunay surface for t > small enough.Hence, the point of our paper is not to show that the ends of the perturbed unduloidfamily are embedded (which is what [7] does), but that all the immersions of this family areembedded on a uniform punctured disk. Equipped with our result, Martin Traizet (in [22]and [21]) showed for the first time how the DPW method can be used to both constructCMC n -noids without symmetries and prove that they are Alexandrov embedded.The theorem we prove is the following one (definitions and notations are clarified inSection 1): Theorem 1.
Let Φ t be a holomorphic frame arising from a perturbed Delaunay potential ξ t defined on a punctured neighbourhood of z = 0 . Suppose that Φ (1 , λ ) = I and that themonodromy of Φ t is unitary. Then, if f t denotes the immersion obtained via the DPWmethod, There exists a family f D t of Delaunay immersions such that for all α < and | t | small enough, k f t ( z ) − f D t ( z ) k R ≤ C α | t || z | α on a uniform neighbourhood of z = 0 .• If t > is small enough, then f t is an embedding of a uniform neighbourhood of z = 0 .• The limit axis of f D t as t tends to can be made explicit. An outline of the proof is given in Section 1.9, together with an explanation of whythe convergence of t to forbids us from using several key results of [7]. Our maps will often depend on a spectral parameter λ that can be in one of the followingsubsets of C ( R > ): D R = { λ ∈ C , | λ | < R } , A R = (cid:8) λ ∈ C , R < | λ | < R (cid:9) , D = { λ ∈ C , | λ | < } , A = { λ ∈ C , | λ | = 1 } . For the coordinate z , we will note ( ǫ > ): D ǫ = { z ∈ C , | z | < ǫ } , S ǫ = { z ∈ C , | z | = ǫ } . Let us define the following (untwisted) loop groups and algebras:•
ΛSL C is the set of smooth maps Φ : A −→ SL C .• ΛSU ⊂ ΛSL C is the set of maps F ∈ ΛSL C such that F ( λ ) ∈ SU for all λ ∈ A .• Λ + SL C ⊂ ΛSL C is the set of maps G ∈ ΛSL C that can be holomorphicallyextended to D and such that G (0) is upper triangular.• Λ R + SL C ⊂ Λ + SL C is the set of maps B ∈ Λ + SL C such that B (0) has positiveelements on the diagonal.• Λ sl C is the set of smooth maps A : A −→ sl C .3 Λ su is the set of maps m ∈ Λ sl C such that m ( λ ) ∈ su for all λ ∈ A .• Λ + sl C ⊂ Λ sl C is the set of maps g ∈ Λ sl C that can be holomorphically extendedto D and such that g (0) is upper triangular.• Λ R + sl C ⊂ Λ + sl C is the set of maps b ∈ Λ + sl C such that b (0) has real elements onthe diagonal.We also use the following notation: O ( t α , z β , λ γ ) = t α z β λ γ f ( t, z, λ ) where f , on its domain of definition, is continuous with respect to ( t, z, λ ) and holomorphicwith respect to ( z, λ ) for any t . If one variable is not specified, its exponent is assumedto be .One step of the DPW method relies on the following Iwasawa decomposition (Theorem8.1.1. in [14]): Theorem 2 (Iwasawa decomposition) . Any element Φ ∈ ΛSL C can be uniquely fac-torised into a product Φ = F × B where F ∈ ΛSU and B ∈ Λ R + SL C . Moreover, the map ΛSL C −→ ΛSU × Λ R + SL C is a C ∞ diffeomorphism for the intersection of the C k topologies (see [7]). The Iwasawa decomposition of a map Φ will often be written: Φ = Uni (Φ) × Pos (Φ) , where Uni (Φ) is called “the unitary factor” of Φ and Pos (Φ) is “the positive factor” of Φ .Using Corollary 3 of Appendix A, note that if Φ is holomorphic on A R , then its unitaryfactor holomorphically extends to A R and its positive factor holomorphically extends to D R . su model of R In the DPW method, immersions are given in a matrix model. The euclidean space R isthus identified with the Lie algebra su by x = ( x , x , x ) ≃ X = − i (cid:18) − x x + ix x − ix x (cid:19) . R identified as su is denoted ( e , e , e ) . In this model, theeuclidean norm is given by k x k = 4 det( X ) . (1)Linear isometries are represented by the conjugacy action of SU on su : H · X = HXH − . The DPW method takes for input data:• A Riemann surface Σ ;• A Λ sl C -valued holomorphic 1-form ξ = ξ ( z, λ ) on Σ called “the DPW potential”which extends meromorphically to D with a pole only at λ = 0 , and which mustbe of the form ξ ( z, λ ) = ∞ X j = − ξ j ( z ) λ j where each matrix ξ j ( z ) depends holomorphically on z and all the entries of ξ − ( z ) are zero except for the upper right entry which must never vanish;• A base point z ∈ Σ ;• An initial condition Φ z ∈ ΛSL C .Given such data, here are the three steps of the DPW method for constructing CMC-1surfaces in R (in the untwisted setting):1. Solve for Φ the Cauchy problem with parameter λ ∈ A : (cid:26) d z Φ( z, λ ) = Φ( z, λ ) ξ ( z, λ ) , Φ( z , λ ) = Φ z ( λ ) . The solution Φ( z, · ) ∈ ΛSL C is called the “holomorphic frame” of the surface. Ingeneral, Φ( · , λ ) is only defined on the universal cover e Σ of Σ (see Section 1.6). Notethat if ξ ( z, · ) can be holomorphically extended to A R ( R > ), then Φ( z, · ) can alsobe holomorphically extended to A R provided that Φ z is holomorphic on A R .2. For all z ∈ e Σ , Iwasawa decompose Φ( z, λ ) = F ( z, λ ) B ( z, λ ) . The decompositionis done pointwise in z , but F ( z, λ ) and B ( z, λ ) depend real-analytically on z . Themap F is called the “unitary frame” of the surface.5. Define f : e Σ −→ su by the Sym-Bobenko formula: f ( z ) = Sym( F ) = i ∂F∂λ ( z, F ( z, − . The map f is then a conformal CMC-1 immersion whose normal map is given by N ( z ) = − i F ( z, (cid:18) − (cid:19) F ( z, − . (2)Its metric and Hopf differential are ds = 2 ρ | ξ − || dz | ,Q = − ξ − ξ dz where ξ klj is the ( k, l ) -entry of the matrix ξ j ( z ) and ρ is the upper-left entry of B ( z, .The theory states that every conformal CMC-1 immersion can be obtained this way. Let ξ be a DPW potential and Φ ∈ ΛSL C a solution of d Φ = Φ ξ . Take a loop H ∈ ΛSU that does not depend on z . Then e Φ = H Φ also satisfies d e Φ = e Φ ξ and gives rise to a rigidmotion of the original surface given by Φ . Let f = Sym ◦ Uni(Φ) and e f = Sym ◦ Uni( e Φ) .Then, e f ( z ) = H (1) · f ( z ) + Sym( H ) . This enjoins us to extend the action of section 1.2 to affine isometries by H ( λ ) · X = H (1) XH (1) − + i ∂H∂λ (1) H (1) − . Note that
ΛSU also acts on the tangent bundle of R via: H · ( p, ~v ) = ( H · p, H (1) · ~v ) . (3)This action will be useful to follow the axis of our surfaces: oriented affine lines aregenerated by pairs ( p, ~v ) and the action of ΛSU on a given oriented affine line correspondsto the action (3) on its generators. 6 .5 Gauging Let (Σ , ξ, z , Φ z ) be a set of DPW data with d Φ = Φ ξ . Let G ( z, λ ) be a holomorphic mapwith respect to z ∈ Σ such that G ( z, · ) ∈ Λ + SL C (such a map is called an “admissiblegauge”). If we define e Φ = Φ G , then Φ and e Φ give rise to the same immersion f . Thisoperation is called “gauging” and one can retrieve e Φ by applying the DPW method to thedata (Σ , ξ · G, z , Φ z G ( z , · )) where ξ · G = G − ξG + G − dG is the action of gauges on potentials. Since Φ is defined as the solution of a Cauchy problem on Σ , it is only defined on theuniversal cover e Σ of Σ . For any deck transformation τ of e Σ , we define the monodromymatrix M τ (Φ) ∈ ΛSL C as follow: Φ( τ ( z ) , λ ) = M τ (Φ)( λ )Φ( z, λ ) . Note that M τ (Φ) does not depend on z . The standard sufficient condition for the immer-sion f to be be well-defined on Σ is the following set of equations, called the monodromyproblem in R : M τ (Φ) ∈ ΛSU , ( i ) M τ (Φ)(1) = ± I , ( ii ) ∂∂λ M τ (Φ)(1) = 0 . ( iii ) Remark 1.
In this paper, the Riemann surface Σ will always be a punctured neighbourhood D ∗ ǫ of z = 0 . Thus, all the deck transformations τ will be associated to a closed loop around z = 0 and we will write M (Φ) instead of M τ (Φ) . Remark 2.
Let
Φ : C ∗ −→ ΛSL C such that M (Φ) ∈ ΛSU . Let e Φ = H ( h ∗ Φ) · G where H ∈ ΛSL C , G is holomorphic at z = 0 and h is a Möbius transformation that leaves z = 0 invariant. Then M ( e Φ) = H M (Φ) H − . Thus, if the monodromy problem for Φ is solved, a sufficient condition for the monodromyproblem for e Φ to be solved is that H ∈ ΛSU . .7 The Delaunay family Delaunay surfaces come in a one-parameter family: for all t ∈ (cid:0) −∞ , (cid:3) \ { } , there existsa unique Delaunay surface, whose weight (as defined in [5]) is πt . The DPW methodcan retrieve these surfaces using the following data: Σ = C ∗ , ξ t ( z, λ ) = A t ( λ ) z − dz, z = 1 , Φ z = I , where A t ( λ ) = (cid:18) rλ − + srλ + s (cid:19) and r, s are functions of t ∈ (cid:0) −∞ , (cid:3) satisfying r, s ∈ R ,r + s = ,rs = t. (4)Note that the system (4) admits two solutions, whether r ≥ s or r ≤ s . For a fixed valueof t , these two solutions give two different parametrisations of the same surface (up to atranslation). If r ≥ s , the unit circle of C ∗ is mapped onto a parallel circle of maximalradius: a bulge of the Delaunay surface. If r ≤ s , the unit circle of C ∗ is mapped ontoa parallel circle of minimal radius: a neck of the Delaunay surface. As t tends to andin the case r ≥ s , the immersions tend towards the parametrisation of a sphere on everycompact subset of C ∗ , which is why we call this setting the “spherical case”. On the otherhand, when r ≤ s and t tends to , the immersions degenerate into a point on everycompact subset of C ∗ . Nevertheless, we call this setting the “catenoidal case” becauseapplying a blowup to the immersions makes them converge towards a catenoid on everycompact subset of C ∗ (see [21] for further details).In any case, the corresponding holomorphic frame is explicit: Φ t ( z, λ ) = z A t ( λ ) as is its monodromy around z = 0 : M (Φ t ) ( λ ) = exp (2 iπA t ( λ )) = cos (2 πµ t ( λ )) I + i sin (2 πµ t ( λ )) µ t ( λ ) A t ( λ ) (5)where µ t ( λ ) = − det A t ( λ ) = 14 + tλ − ( λ − . (6)Note that the conditions (4) have been chosen in order for the monodromy problem ofSection 1.6 to be solved. The axis of the surface is given by { ( x, , − r ) , x ∈ R } and itsweight is πt . Thus, the induced surface is an unduloid if t > and a nodoid if t < .8 emark 3. In order to deal with a single-valued square root of µ t ( λ ) and to avoid someresonance cases in Section 3, we set T > and R > small enough for (cid:12)(cid:12)(cid:12)(cid:12) µ t ( λ ) − (cid:12)(cid:12)(cid:12)(cid:12) < to hold for all ( t, λ ) ∈ ( − T, T ) × A R . We take a Delaunay potentials family as in section 1.7 and we perturb it for z in a smalluniform neighbourhood of : Definition 1 (Perturbed Delaunay potential) . Let ǫ > . A perturbed Delaunay potentialis a one-parameter family { ξ t } t ∈ ( − T,T ) of DPW potentials, holomorphic on D ∗ ǫ × A R andof the form ξ t ( z, λ ) = A t ( λ ) z − dz + R t ( z, λ ) dz where A t is a Delaunay residue as in Section 1.7 and R t ( z, λ ) ∈ C with respect to ( t, z, λ ) ,is holomorphic on D ǫ × A R for all t and satisfies R ( z, λ ) = 0 . The following set of hypotheses will be used to make sure that our holomorphic frameshave a C regularity, are holomorphic with respect to ( z, λ ) , and solve the monodromyproblem: Hypotheses 1.
Let ξ t be a perturbed Delaunay potential. Let Φ t be a holomorphic frameassociated to it. We suppose that• For some t ∈ ( − T, T ) and z ∈ D ∗ ǫ , Φ t ( z, · ) is holomorphic on A R ,• Φ t ( z, λ ) is continous with respect to ( t, z, λ ) ,• The monodromy is unitary: M (Φ t ) ∈ ΛSU . Remark 4.
When needed, one can replace
R > by a smaller value in order for Φ t tobe holomorphic on A R and continuous on A R . The theorem we prove in this paper is the following:
Theorem 3.
Let ξ t be a perturbed Delaunay potential and Φ t a holomorphic frame asso-ciated to ξ t satisfying Hypotheses 1 and such that Φ (1 , λ ) = I . Let f t = Sym (Uni(Φ t )) .Then, . For all α < there exist constants ǫ > , T > and C > such that for all < | z | < ǫ and | t | < T , k f t ( z ) − f D t ( z ) k R ≤ C | t || z | α where f D t is a Delaunay immersion of weight πt .2. There exist T ′ > and ǫ ′ > such that for all < t < T ′ , f t is an embedding of { < | z | < ǫ ′ } .3. If r ≥ s , the limit axis as t tends to of f D t is the oriented line generated by ( − e , − ~e ) .If r ≤ s , the limit axis as t tends to of f D t is the oriented line generated by (0 , − ~e ) . Remark 5.
We do not have to assume that ∈ D ǫ for Φ to be defined at z = 1 . Thisonly comes from the fact that ξ is defined on C ∗ , which implies that Φ is defined on theuniversal cover f C ∗ . In Section 3 we start the proof of Theorem 3 by gauging the potential and changingcoordinates. Starting from ξ t = A t z − dz + O ( t, z ) dz we gain an order on z and obtain the following new potential: e ξ t = A t z − dz + O ( t, z ) dz. We then use the Fröbenius method and the new holomorphic frame is e Φ t = f M t z A t (cid:0) I + O ( t, z ) (cid:1) . In Section 4, we use this estimate on e Φ t to prove the convergence of the immersions: (cid:13)(cid:13)(cid:13) e f t ( z ) − e f D t ( z ) (cid:13)(cid:13)(cid:13) R ≤ C | t || z | α , α < where e f D t is a Delaunay immersion whose axis can be explicitly computed. To do so, weneed to know the asymptotic behaviour of the positive part Pos( e Φ t ) , which we computeusing the fact that e f D t ( C ∗ ) is a surface of revolution.10inally, Section 5 proves that perturbations of unduloids are embedded on a uniformneighbourhood of the origin.Although the method of this paper is inspired by what Kilian, Rossman and Schmittdid in [7], their results cannot be used to prove our theorem. This is mainly because theasymptotics given in [7] for a fixed value of our parameter t do not hold as t tends to .As an example, consider the proof of Lemma 2.5 in [7]: with our hypotheses, the constantthey call κ becomes a function of t such that (with our notation of Section 3.2) κ | t =0 = c (0 , = 0 . Later in the proof, computing the determinant of the linear map L gives det L = O ( t ) and their gauged potential is then of the form b ξ t = A t z − dz + O ( t − , z ) dz, the corresponding holomorphic frame being b Φ t = c M t z A t (cid:0) I + O ( t − , z ) (cid:1) . Applying the Sym-Bobenko formula would give at best (cid:13)(cid:13)(cid:13) b f t ( z ) − b f D t ( z ) (cid:13)(cid:13)(cid:13) R ≤ C | t | | z | α , α < (7)which is not enough to show the convergence of the immersions on the compact sets of C ∗ as t tends to . Note that gaining one order on | t | in the estimate (7) is still not enoughto show the embeddedness of b f t , since the first catenoidal neck of b f D t , which has a size ofthe order of t , is attained for | z | ∼ | t | as t tends to .Finally, some bounds used in [7] such as (see Lemma 1.11 in [7]) c ( λ ) = max x ∈ [0 ,ρ ) k B ( x, λ ) k depend on t in our framework and may explode as t tends to .11 An application
Before proving Theorem 3, we must take account of the fact that one of its hypotheses istoo restrictive. Indeed, Φ (1 , λ ) = I has no reason to hold when one wants to constructexamples, as Martin Traizet did in [22] and [21]. We thus show here on a specific examplehow to ensure this hypothesis by gauging the potential and changing coordinates.In all the section, ξ t is a perturbed Delaunay potential with r ≥ s and Φ t is a holo-morphic frame associated to ξ t , satisfying Hypotheses 1 and such that Φ (1 , λ ) = M ( λ ) where M ( λ ) = (cid:18) a bλ − cλ d (cid:19) ∈ ΛSL C ( a, b, c, d ∈ C ) . (8)After some simplification, we will be able to apply Theorem 3 even though Φ (1 , λ ) =I . The only difference in the conclusion will be in the third point: the limit axis as t tendsto of the model Delaunay surface f D t will be the oriented line generated by Q · (0 , ~e ) where Q = Uni [ M H ] (9)with H ( λ ) = 1 √ (cid:18) − λ − λ (cid:19) . (10)The method involves gauging, changing coordinates and applying an isometry, and relieson the fact that one can explicitly compute the Iwasawa decomposition of M H . Indeed,for all a, b, c, d ∈ C such that ad − bc = 1 , (cid:18) a bλ − cλ d (cid:19) = 1 p | b | + | d | (cid:18) d bλ − − bλ d (cid:19) × p | b | + | d | (cid:18) (cid:0) ab + cd (cid:1) λ | b | + | d | (cid:19) (11)is the Iwasawa decomposition of the left-hand side term. Note that if the matrix M isexplicit, then this formula makes both the matrix Q in Equation (9) and the limit axis of f D t explicit because M H and M have the same form. Lemma 1.
Let ξ t be a perturbed Delaunay potential as in Definition 1 with r ≥ s . Let Φ t be a holomorphic frame associated to it, satisfying Hypotheses 1 and such that Φ (1 , λ ) = M ( λ ) as in (8) . Then there exists a Möbius transformation that leaves z = 0 invariantand a gauge G such that:1. the new potential e ξ t = ( h ∗ ξ t ) · G is also a perturbed Delaunay potential with the sameresidue than ξ t , . the holomorphic frame e Φ t associated to e ξ t satisfies Hypotheses 1 with e Φ (1 , λ ) ∈ ΛSU .Proof. Let A t and R t be as in Definition 1. Then e ξ t = G − (cid:0) A t h − dh + ( h ∗ R t ) dh (cid:1) G + G − dG. The Möbius transformation we are looking for satisfies h (0) = 0 and thus h − dh = z − dz + O ( z ) dz. Wanting e ξ t to have a simple pole at z = 0 , we look for a gauge G that is holomorphicat z = 0 . Wanting the residue of e ξ t to be A t , we suppose that G (0 , λ ) = I . These twoconditions together with e ξ = A z − dz enjoin us to solve the following Cauchy problem: (cid:26) dG = GA z − dz − A Gh − dhG (0) = I . (12)If we write h ( z ) = zpz + q , p ∈ C , q ∈ C ∗ , then the only solution of (12) is given (by Maple) by: G ( z, λ ) = q qpz + q λpz √ q ( pz + q ) q pz + qq and a straightforward computation allows us to check that G satisfies (12). Setting <ǫ ′ < ǫ with ǫ ′ < | q || p | if necessary, this proves the first point of the lemma.In order to prove the second point, diagonalise A = HDH − with H as in (10) andcompute e Φ (1 , λ ) = M ( λ ) H ( λ ) (cid:0) h (1) D H ( λ ) − G (1 , λ ) H ( λ ) (cid:1) H ( λ ) − (13)where D = (cid:18) − (cid:19) . Hence e Φ (1 , · ) is holomorphic on A R . Moreover, the fact that e ξ t is C in ( t, z, λ ) togetherwith remark 2 imply that e Φ t satisfies Hypotheses 1. Finally, compute h (1) D H ( λ ) − G (1 , λ ) H ( λ ) = √ q λ p √ q √ q ! Pos (
M H ) = (cid:18) ρ λµ ρ − (cid:19) where ρ = √ p | b − a | + | d − c | , µ = 1 √ × ( a + b )(¯ b − ¯ a ) + ( c + d )( ¯ d − ¯ c ) p | b − a | + | d − c | . Then, setting p = − ρµ, q = ρ , Equation (13) becomes ( Q is defined in (9)) e Φ (1 , λ ) = QH − ∈ ΛSU because H ∈ ΛSU .If one wants to apply Theorem 3, it then suffices to set b Φ t = HQ − e Φ t where e Φ t is constructed by Lemma 1. Let b f D t be the model Delaunay immersion towardswhich the immmersion Sym (cid:16)
Uni( b Φ t ) (cid:17) converges. Theorem 3 then states that the limitaxis as t tends to of b f D t is the oriented line generated by ( − e , − ~e ) . Compute H − · ( − e , − ~e ) = ( − e , ~e ) ≃ (0 , ~e ) to prove that Sym (Uni(Φ t )) converges to a model Delaunay surface whose limit axis as t tends to is Q · (0 , ~e ) . The following corollary summarises this section: Corollary 1.
Let ξ t be a perturbed Delaunay potential with r ≥ s and Φ t a holomorphicframe associated to ξ t satisfying Hypotheses 1 and such that Φ (1 , λ ) is of the form givenby (8) . Let f t = Sym (Uni(Φ t )) . Then,1. For all α < there exist constants ǫ > , T > and C > such that for all < | z | < ǫ and | t | < T , k f t ( z ) − f D t ( z ) k R ≤ C | t || z | α where f D t is a Delaunay immersion of weight πt .2. There exist T ′ > and ǫ ′ > such that for all < t < T ′ , f t is an embedding of { < | z | < ǫ ′ } .3. The limit axis as t tends to of f D t is the oriented line generated by Q · (0 , ~e ) where Q is given by Equation (9) . The z A P form of Φ t Let us start the proof of Theorem 3: let ξ t be a perturbed Delaunay potential and Φ t aholomorphic frame associated to ξ t satisfying Hypotheses 1 and such that Φ (1 , λ ) = I .In this section, we want to apply the Fröbenius method and write Φ t in a z A P form.Unfortunately, the underlying Fuchsian system seems to admit resonance points. Ourgoal is to avoid them and to gain an order of convergence in the matrix P of the z A P form. We will obtain the following result: Proposition 1.
There exist a change of coordinate h t and a gauge G t such that, denoting e Φ t = h ∗ t (Φ t G t ) and e ξ t = h ∗ t ( ξ t · G t ) , e ξ t is a perturbed Delaunay potential and e Φ t is a holomorphic frame associated to e ξ t satis-fying Hypotheses 1 and such that e Φ (1 , λ ) = I . Moreover, e Φ t ( z, λ ) = f M t ( λ ) z A t ( λ ) e P t ( z, λ ) (14) where f M t ∈ ΛSL C is continuous and holomorphic on A R for all t and e P t : D ǫ ′ −→ ΛSL C is C , holomorphic on D ′ ǫ × A R for all t and satisfies e P t ( z, λ ) = I + O ( t, z ) . In this section, we use the Fröbenius method to write Φ t in a z A P form, and extend thisform to the resonance points. We will thus prove: Proposition 2.
There exist M t ∈ ΛSL C continuous and holomorphic on A R for all t and P t : D ǫ −→ ΛSL C continuous and holomorphic on D ǫ × A R for all t satisfying P t (0 , λ ) = I and Φ t ( z, λ ) = M t ( λ ) z A t ( λ ) P t ( z, λ ) . Let us first recall the Fröbenius method in the non-resonant case (see [19] and [18]).Let ǫ > and ξ be a holomorphic -form from D ∗ ǫ to M ( C ) defined by ξ ( z ) = Az − dz + X k ∈ N C k z k dz. k ∈ N , let P k solve ( P = I , L k +1 ( P k +1 ) = P i + j = k P i C j (15)where for all n ∈ N , L n : M ( C ) −→ M ( C ) X [ A, X ] + nX.
Then P ( z ) = P k ∈ N P k z k is holomorphic on D ǫ and Φ( z ) = z A P ( z ) is holomorphic on theuniversal cover f D ∗ ǫ of D ∗ ǫ and solves d Φ = Φ ξ .Let us now recall Lemma 2.2 of [7] in our framework: Lemma 2.
Let A ∈ sl C such that A = µ I . Then for all n ∈ N , det L n = n (cid:0) n − µ (cid:1) (16) and L − n ( X ) = 1 n (cid:18) X − n − µ ( n I − A ) [ A, X ] (cid:19) (17)Corollary 2 follows from Remark 3 and Equation (16). Corollary 2.
Let L t,n ( X ) = [ A t ( λ ) , X ] + nX .• For all n ≥ , L t,n is invertible on ( t, λ ) ∈ ( − T, T ) × D ∗ R .• For n = 1 , L t, is invertible on ( t, λ ) ∈ ( − T, T ) \{ } × D ∗ R \{ } . Remark 6.
If we use the Ansatz given by the Fröbenius method and write Φ t ( z, λ ) = M t ( λ ) z A t ( λ ) P t ( z, λ ) (18) where P t ( z, λ ) = ∞ X k =0 P t,k ( λ ) z k , note that the resonance points only occur in the computation of P t, ( λ ) because L t,n isinvertible on ( t, λ ) ∈ ( − T, T ) × A R for all n ≥ . Thus, we only need to extend P t, ( λ ) at t = 0 and λ = 1 to extend the z A P form of Φ t . According to (15) , P t, ( λ ) = L − t, ( tC t ( λ )) (19) and the form of det L t, shows that P t, has at most a pole of order at λ = 1 . Moreover, det L t, = O ( t ) and tC t = O ( t ) , so we already know that P t (and as a consequence, M t )extends to t = 0 .
16t remains to extend the z A P form (18) to λ = 1 . To do this, we adapt the techniquesused in Lemma 2.5 of [15] to prove the following unitary × commutator lemma: Lemma 3.
Let M : A R \{ } −→ SL C holomorphic on A R \{ } with at most a poleat λ = 1 . Let t = 0 , Q = exp (2 iπA t ) ∈ ΛSU and suppose that for all λ ∈ A \{ } , M Q M − ∈ SU . Then there exist U ∈ ΛSU and K : A R \{ } −→ SL C holomorphicsuch that (cid:26) M = U K [ A t , K ] = 0 . Proof.
We first apply Lemma 2.5 of [15] to construct U and K satisfying M = U K and [ Q , K ] = 0 on A \{ } . The map U is holomorphic on a small neighbourhood of A .Without loss of generality, let this neighbourhood be A R . Then, K is meromorphic on A R \{ } with at most a pole at λ = 1 . Hence the map λ [ Q ( λ ) , K ( λ )] is holomorphicon A R \{ } and vanishes on A \{ } . Thus, for all λ ∈ A R \{ } , [ Q ( λ ) , K ( λ )] = 0 . (20)Recalling Equation (5), Q = cos(2 πµ t )I + i sin(2 πµ t ) µ t A t . Hence Equation (20) implies that [ A t , K ] = 0 wherever µ t ( λ ) = . Using (6), [ A t ( λ ) , K ( λ )] =0 for all ( t, λ ) ∈ ( − T, T ) \{ } × A R \{ } .We can now extend the z A P form of Φ t to λ = 1 . For t = 0 and λ ∈ A \{ } , useLemma 3 to write Φ t ( z, λ ) = U t ( λ ) z A t ( λ ) K t ( λ ) P t ( z, λ ) . Let ǫ > small enough for P t ( · , λ ) to be defined on D ǫ . On S ǫ × A \{ } , Φ t and z A t arebounded. Then the map ( z, λ ) K t P t is bounded on S ǫ × A \{ } and holomorphic on D ǫ × A \{ } , so it is bounded on D ǫ × A \{ } . But P t (0 , λ ) = I , so K t is bounded on A \{ } . Thus, P t is bounded on D ǫ × A \{ } . But P t is holomorphic on D ǫ × A R \{ } with at most a pole at λ = 1 , so P t is holomorphic on D ǫ × A R and M t is holomorphic on A R . This ends the proof of Proposition 2. ξ t The fact that there exists a holomorphic frame Φ t associated to ξ t such that M (Φ t ) ∈ ΛSU and Φ (1 , λ ) = I gives us a piece of information on the potential ξ t . Let C t ( λ ) ∈ sl C so that ξ t ( z, λ ) = A t ( λ ) z − dz + tC t ( λ ) dz + O ( t, z ) dz C t ( λ ) = (cid:18) c ( t, λ ) λ − c ( t, λ ) c ( t, λ ) − c ( t, λ ) (cid:19) . (21)Define p t = sc ( t,
0) + rc ( t, . (22) Lemma 4.
The quantity p t vanishes at t = 0 .Proof. First, note that Φ (1 , λ ) = I implies that Φ ( z, λ ) = z A ( λ ) , and thus M (Φ ) = − I . Let γ ⊂ D ∗ ǫ be a closed loop around . Apply Proposition 5 of Appendix B to get( X ′ denotes the derivative of X at t = 0 and R t is the holomorphic part of ξ t ) M (Φ t ) ′ = Z γ z A ξ ′ z − A × M (Φ )= − Z γ z A (cid:0) A ′ z − (cid:1) z − A dz − Z γ z A R ′ z − A dz = M ( z A t ) ′ − Z γ z A R ′ z − A dz. But M (Φ t ) , M ( z A t ) ∈ ΛSU and M (Φ ) = M ( z A ) = − I . Thus, M (Φ t ) ′ , M ( z A t ) ′ ∈ Λ su and Z γ z A R ′ z − A dz ∈ Λ su . (23)Diagonalise A = HDH − with D = (cid:18) − (cid:19) and H ∈ ΛSU to be expressed later. Then z D = 1 √ z (cid:18) z
00 1 (cid:19) and Z γ z A R ′ z − A dz = Z γ Hz D H − ( C + O ( z )) Hz − D H − = H (cid:0) Res z =0 z D H − C Hz − D (cid:1) H − . Equation (23) and H ∈ ΛSU imply that Res z =0 (cid:0) z D H − C Hz − D (cid:1) ∈ Λ su . (24)18enoting by c ( λ ) the bottom-left entry of H − C H and looking at the product z D ( H − C H ) z − D ,Equation (24) gives (cid:18) c ( λ ) 0 (cid:19) ∈ Λ su and thus, c ( λ ) = (cid:0) H − C H (cid:1) ≡ (25)Two cases can occur:• If r ≥ s , H = 1 √ (cid:18) − λ − λ (cid:19) ∈ ΛSU and computation gives c ( λ ) = − λ (cid:18) c (0 , λ ) + c (0 , λ )2 (cid:19) + c (0 , λ )2 . Using Equation (25), c (0 ,
0) = 0 and p = 0 .• If r ≤ s , the same reasoning applies with H ( λ ) = 1 √ (cid:18) −
11 1 (cid:19) and c ( λ ) = − λ − c (0 , λ )2 + c (0 , λ )2 − c (0 , λ ) . Thus, c (0 ,
0) = 0 and p = 0 . We can now prove Proposition 1 by following the method used in Section 2.2 of [7]:gauging the potential. The gauge we will use is of the following form: G t ( z, λ ) = exp ( g t ( λ ) z ) (26)which is an admissible gauge provided that g t ∈ Λ + sl C . This is why we need the followinglemma: Lemma 5.
Let g t ( λ ) = p t A t ( λ ) − P t, ( λ ) where P t, is defined in Equation (19) . Then . The map g t is in Λ + sl C .2. The map g t extends to t = 0 with g = 0 .Proof. To prove the first point, let t = 0 and use Equations (19), (21), (17) and (22) tocompute (this is a tedious calculation) P t, ( λ ) = λ − (cid:18) rp t (cid:19) + λ (cid:18) ⋆ ⋆sp t ⋆ (cid:19) + O ( λ ) . Thus, g t ( λ ) = p t A t ( λ ) − P t, ( λ ) = λ − (cid:18) (cid:19) + λ (cid:18) ⋆ ⋆ ⋆ (cid:19) + O ( λ ) . For the second point, use Equations (19) and (17) to write for t = 0 : P t, = t L − t, ( C t ) = t (cid:18) C t − − µ t (I − A t ) [ A t , C t ] (cid:19) . Note that C t is continuous at t = 0 because ξ t ∈ C and that − µ t = O ( t ) to extend P t, to t = 0 . Moreover, recall Lemma 4, Equation (6) and diagonalise A = HDH − toget: g = − λ λ − H (I − D ) (cid:2) D, H − C H (cid:3) H − . A straightforward computation gives (I − D ) (cid:2) D, H − C H (cid:3) = (cid:18) − c ( λ ) 0 (cid:19) with c ( λ ) as in Equation (25). Hence g = 0 .Let G t be the gauge defined by (26). Then the gauged potential has the form ξ t · G t ( z, λ ) = A t ( λ ) z − dz + ([ A t ( λ ) , g t ( λ )] + g t ( λ ) + tC t ( λ )) dz + O ( t, z ) dz + O ( g t z ) dz = A t ( λ ) z − dz + ( L t, ( g t ( λ )) + tC t ( λ )) dz + O ( t, z ) dz = A t ( λ ) z − dz + p t A t ( λ ) dz + O ( t, z ) dz, because of Equation (19). This gauge has been chosen to fit with the following change ofcoordinate: h t ( z ) = z p t z . e ξ t = A t dz p t z + p t A t dz (1 + p t z ) + O ( t, z ) dz = A t z − dz + O ( t, z ) dz because p = 0 . Apply the Fröbenius method to e ξ t to obtain (14) and choose ǫ ′ ≤ ǫ suchthat for all t = 0 , ǫ ′ < | p t | − to end the proof of Proposition 1. In this section, we prove the first and third points of Theorem 3. In the end, we want tocompare Φ t ( z, λ ) = M t ( λ ) z A t ( λ ) (I + O ( t, z )) to Φ D t ( z, λ ) = M t ( λ ) z A t ( λ ) . We will denote F D t = Uni(Φ Dt ) and f D t = Sym( F D t ) . We first want to make sure that Φ D t induces a Delaunay surface for all t . For this purpose,recall Lemma 1.12 in [7], which implies that f D t is a Delaunay surface of weight πt . Hence,there exists a rigid motion φ of R such that φ ◦ f D t has the following parametrisation: φ ◦ f D t : Σ −→ R z = e x + iy ( τ t ( x ) , σ t ( x ) cos y, σ t ( x ) sin y ) where ( τ t ( x ) , σ t ( x )) is the profile curve of the surface. Recalling that the coordinates areisothermal gives the following metric: ds t = σ t | dz | | z | . (27)Let us compare the asymptotic behaviours of the unitary parts of Φ t and Φ D t for λ ∈ A using, as in [7], a Cauchy formula. We will use the following norms:• For v = ( v , v ) ∈ C , | v | = ( | v | + | v | ) .• For M ∈ M ( C ) , k M k = sup | v | =1 | M v | . 21 For Ψ :
E −→ M ( C ) , k Φ k E = sup λ ∈E k Ψ( λ ) k . Lemma 6.
For all α < there exist constants ǫ > , T > and C > such that for all < | z | < ǫ and | t | < T , (cid:13)(cid:13)(cid:13)(cid:0) F D t (cid:1) − F t − I (cid:13)(cid:13)(cid:13) A ≤ C | t || z | α (28) and (cid:13)(cid:13)(cid:13)(cid:13) ∂∂λ h(cid:0) F D t (cid:1) − F t i(cid:13)(cid:13)(cid:13)(cid:13) A ≤ C | t || z | α . (29) Proof.
The first step is to estimate the norm of the positive part B D t of Φ D t . We firstestimate Φ D t for | z | < : noting that A t is diagonalisable, that its eigenvalues tend to ± / as t → , and recalling that M t is continuous at t = 0 ensure that for all α < there exists ( T, R ) and C > such that for all | t | < T , (cid:13)(cid:13) Φ D t ( z, λ ) (cid:13)(cid:13) A R ≤ C | z | − − − α . We then estimate F D t : let γ ⊂ C ∗ be a path from z to , use Equation (39) of AppendixC and Equation (27) to get (cid:13)(cid:13) F D t ( z, λ ) (cid:13)(cid:13) A R ≤ C (cid:13)(cid:13) F D t (1 , λ ) (cid:13)(cid:13) A R × exp (cid:18) ( R − Z γ | σ t (log | z | ) || z | (cid:19) . But σ t is uniformly bounded because so is the distance between the profile curve and theaxis of a Delaunay surface. Moreover, the unitary frame at z = 1 is also bounded. Hencethe existence, for R > small enough, of a constant C ≥ such that (cid:13)(cid:13) F D t ( z, λ ) (cid:13)(cid:13) A R ≤ C | z | − − α . We can now estimate the positive factor: for all α < there exist T > , R > and C ≥ such that for all | t | < T and | z | < (cid:13)(cid:13) B D t ( z, λ ) (cid:13)(cid:13) A R ≤ (cid:13)(cid:13) F D t ( z, λ ) − (cid:13)(cid:13) A R × (cid:13)(cid:13) Φ D t ( z, λ ) (cid:13)(cid:13) A R ≤ C | z | α − . We then define e Φ t := (cid:16)(cid:0) F D t (cid:1) − F t (cid:17) × (cid:16) B t (cid:0) B D t (cid:1) − (cid:17) = B D t (cid:0) Φ D t (cid:1) − Φ t (cid:0) B D t (cid:1) − =: e F t × e B t e F t ∈ ΛSU and e B t ∈ Λ R + SL C and thus have (cid:13)(cid:13)(cid:13)e Φ t ( z, λ ) − I (cid:13)(cid:13)(cid:13) A R = (cid:13)(cid:13)(cid:13) B D t ( z, λ ) ( P t ( z, λ ) − I ) (cid:0) B D t ( z, λ ) (cid:1) − (cid:13)(cid:13)(cid:13) A R ≤ (cid:13)(cid:13) B D t ( z, λ ) (cid:13)(cid:13) A R O ( t, | z | ) ≤ C | t || z | α . Let n k denote the seminorms n k ( X ) = k X j =0 (cid:13)(cid:13)(cid:13)(cid:13) ∂ k X∂λ k (cid:13)(cid:13)(cid:13)(cid:13) A . Apply Cauchy formula with λ ∈ ∂ A R to get n k (cid:16)e Φ t − I (cid:17) ≤ c k | t || z | α , ∀ k ∈ N where c k > are uniform constants. But Uni( e Φ t ) = e F t = (cid:0) F D t (cid:1) − F t and Iwasawadecomposition is a C -diffeomorphism, so n (cid:16) e F t − I (cid:17) ≤ C | t || z | α and n (cid:16) e F t − I (cid:17) ≤ C | t || z | α . We then have (28) and (29).The asymptotic behaviour of ∂ e F t ∂λ allows us to prove the convergence of immersions asstated in the first point of Theorem 3. The Sym-Bobenko formula for R implies that (weomit the index t ) iF ( z, ∂ ( F − F D ) ∂λ ( z, F D ( z, − = i ∂F D ∂λ ( z, F D ( z, − − i ∂F∂λ ( z, F ( z, − = f D ( z ) − f ( z ) . We can then compute (cid:13)(cid:13) f t ( z ) − f D t ( z ) (cid:13)(cid:13) R = 4 det (cid:0) f t ( z ) − f D t ( z ) (cid:1) = − ∂ ( F − t F D t ) ∂λ ( z, ≤ C t | z | α . And then for all α < there exist constants ǫ > , T > and C > such that for all < | z | < ǫ and | t | < T , k f t ( z ) − f D t ( z ) k R ≤ C | t || z | α . (30)23o prove the third point of Theorem 3, use (4) and note that M = I . So the axis of f D t as t → is the same that the axis of the unperturbed Delaunay surface induced by z A t .In order to prove that the surface is embedded, we will need the convergence of thenormal maps: Proposition 3.
For all α < there exist constants ǫ > , T > and C > such thatfor all < | z | < ǫ and | t | < T , (cid:13)(cid:13) N t ( z ) − N D t ( z ) (cid:13)(cid:13) R ≤ C | t || z | α Proof.
Use the definition of the normal maps in Equation (2) to write N t ( z ) − N D t ( z ) = − i F D t ( z, h AM e A + AM + M e A i F D t ( z, − where A = F D t ( z, − F t ( z, − I = O ( t, | z | α ) , e A = F t ( z, − F D t ( z, − I = O ( t, | z | α ) and M = (cid:18) − (cid:19) . Use equation (1) to get the conclusion.It remains to show that the surface is embedded if t > . We suppose in this section that < t < T . The asymptotic behaviour of f t and the factthat f D t is an embedding for all t allow us to show that f t is an embedding of a sufficientlysmall uniform neighbourhood of z = 0 for t small enough. We first give a general resultof embeddedness and then apply this result to show that our surfaces are embedded. Proposition 4.
Let f R n : C ∗ −→ M R n = f R n ( C ∗ ) ⊂ R be a sequence of complete im-mersions with normal maps N R n and an end at z = 0 . Suppose that for all n there exists r n > such that the tubular neighbourhood Tub r n M R n of M R n is embedded. Suppose thatfor all ǫ > there exists < ǫ ′ < ǫ such that for all n ∈ N , x ∈ S ǫ and y ∈ D ∗ ǫ ′ , (cid:13)(cid:13) f R n ( x ) − f R n ( y ) (cid:13)(cid:13) R > r n . (31)24 et U ∗ ⊂ C ∗ be a punctured neighbourhood of z = 0 and f n : U ∗ −→ R a sequence ofimmersions with normal maps N n satisfying sup n ∈ N (cid:13)(cid:13) f n ( z ) − f R n ( z ) (cid:13)(cid:13) R r n −→ z → (32) and sup z ∈ U ∗ (cid:13)(cid:13) N n ( z ) − N R n ( z ) (cid:13)(cid:13) R −→ n →∞ . (33) Then there exist ǫ ′ > and N ∈ N such that for all n ≥ N , f n is an embedding of D ∗ ǫ ′ .Proof. Let us split the proof in several steps.•
Claim : there exists ǫ > such that the map ϕ n : D ∗ ǫ −→ M R n z π n ◦ f n ( z ) (where π n is the projection from Tub r n M R n onto M R n ) is well-defined and satisfies (cid:13)(cid:13) ϕ n ( z ) − f R n ( z ) (cid:13)(cid:13) R < r n (34)for all z ∈ D ∗ ǫ .To prove this first claim, use Hypothesis (32): there exists ǫ > such that for all n ∈ N and z ∈ D ∗ ǫ (cid:13)(cid:13) f n ( z ) − f R n ( z ) (cid:13)(cid:13) R < r n . (35)So f n ( D ∗ ǫ ) ⊂ Tub rn M R n and ϕ n is well-defined. Moreover, using (35) and the triangleinequality, for all z ∈ D ∗ ǫ (cid:13)(cid:13) ϕ n ( z ) − f R n ( z ) (cid:13)(cid:13) R ≤ k ϕ n ( z ) − f n ( z ) k R + (cid:13)(cid:13) f n ( z ) − f R n ( z ) (cid:13)(cid:13) R < r n and Equation (34) holds. We fix ǫ and ǫ ′ so that Equation (31) is satisfied.• Claim 2 : there exists N ∈ N such that for all n ≥ N , ϕ n is a local diffeomorphismon D ∗ ǫ .Let z ∈ D ∗ ǫ . In order to show that ϕ n is a local diffeomorphism, we show that hN ϕ n ( z ) , N n ( z ) i > (36)25here N ϕ n is defined by N ϕ n : D ∗ ǫ −→ S ⊂ R z η R n ( ϕ n ( z )) and η R n is the Gauss map of M R n . First, let γ ⊂ M R n be a path joining ϕ n ( z ) to f R n ( z ) .Using the fact that Tub r n M R n is embedded, one has (cid:13)(cid:13) dη R n (cid:13)(cid:13) ≤ r n and (cid:13)(cid:13) N ϕ n ( z ) − N R n ( z ) (cid:13)(cid:13) R ≤ r n × | γ | . Let σ ( t ) = (1 − t ) f n ( z ) + tf R n ( z ) , t ∈ [0 , . Then, (cid:13)(cid:13) σ ( t ) − f R n ( z ) (cid:13)(cid:13) R ≤ (1 − t ) (cid:13)(cid:13) f n ( z ) − f R n ( z ) (cid:13)(cid:13) R < r n (37)because of Equation (35). Let γ = π n ◦ σ . Note that Equation (37) implies that σ ⊂ Tub rn M R n and restricting π n to Tub rn M R n gives k dπ n k ≤ r n r n − r n = 2 and thus | γ | < r n . Hence, (cid:13)(cid:13) N ϕ n ( z ) − N R n ( z ) (cid:13)(cid:13) < . Use Hypothesis (33) to choose a uniform N ∈ N such that for all n ≥ N , kN ϕ n ( z ) − N n ( z ) k ≤ (cid:13)(cid:13) N ϕ n ( z ) − N R n ( z ) (cid:13)(cid:13) + (cid:13)(cid:13) N R n ( z ) − N n ( z ) (cid:13)(cid:13) < √ , which proves Equation (36) and this second claim. We fix such N and n .• Claim 3 : the restriction e ϕ n : ϕ − n ( ϕ n ( D ∗ ǫ ′ )) ∩ D ∗ ǫ −→ ϕ n ( D ∗ ǫ ′ ) z ϕ n ( z ) is a covering map. 26t sufices to show that e ϕ n is a proper map. Let ( x i ) i ∈ N ⊂ ϕ − n ( ϕ n ( D ∗ ǫ ′ )) ∩ D ∗ ǫ such that ( e ϕ n ( x i )) i ∈ N converges to p ∈ ϕ n ( D ∗ ǫ ′ ) . Then ( x i ) i converges to x ∈ D ǫ . Using Equation(34) and the fact that f R n has an end at , x = 0 . If x ∈ ∂ D ǫ , denoting e x ∈ D ∗ ǫ ′ such that e ϕ n ( e x ) = p , one has (cid:13)(cid:13) f R n ( x ) − f R n ( e x ) (cid:13)(cid:13) R < (cid:13)(cid:13) f R n ( x ) − p (cid:13)(cid:13) R + (cid:13)(cid:13) f R n ( e x ) − e ϕ n ( e x ) (cid:13)(cid:13) R < r n which contradicts the definition of ǫ ′ . Thus, e ϕ n is a proper local diffeomorphism betweenlocally compact spaces, i.e. a covering map.• Claim 4 : this covering map is one-sheeted.To compute the number of sheets, let γ : [0 , −→ D ∗ ǫ ′ be a loop of winding number around , Γ = f R n ( γ ) and e Γ = e ϕ n ( γ ) ⊂ M R n and let us construct a homotopy between Γ and e Γ . Let σ t : [0 , −→ R s (1 − s ) e Γ( t ) + s Γ( t ) . For all t, s ∈ [0 , , k σ t ( s ) − Γ( t ) k R < r n which implies that σ t ( s ) ∈ Tub r n M R n because M R n is complete. One can thus define thefollowing homotopy between Γ and e Γ H : [0 , −→ M R n ( s, t ) π n ◦ σ t ( s ) where π n is the projection from Tub r n M R n to M R n . Using the fact that f R n is an embedding,the degree of Γ is one, and the degree of e Γ is also one. Hence, e ϕ n is one-sheeted.• Conclusion : the map e ϕ is a diffeomorphism, so f n ( D ∗ ǫ ′ ) is a graph over M R n containedin its embedded tubular neighbourhood and f n ( D ∗ ǫ ′ ) is thus embedded.We can now apply Proposition 4 to each case. Let ( t n ) be any sequence in ( − T, T ) such that t n → .• If r ≥ s , we set b f R n = f D t n and b f n = f t n . We aim to apply Proposition 4 on b f R n and b f n . The tubular radius r n is of the order of t n and Hypothesis (31) is satisfiedbecause b f R n tends to an immersion of a sphere. Equation (30) and Proposition 3ensure that Hypotheses (32) and (33) hold.27 If r ≤ s , we set b f R n = t n f D t n and b f n = t n f t n . We aim to apply Proposition 4 on b f R n and b f n . The tubular radius r n is of the order of and Hypothesis (31) is satisfiedbecause b f R n tends to an immersion of a catenoid (see [21]). Equation (30) andProposition 3 ensure that Hypotheses (32) and (33) hold.The second point of our theorem is then proved. A Iwasawa extended
In this section, we note A R , = (cid:8) λ ∈ C : R < | λ | < (cid:9) . Lemma 7.
Let F : A R , −→ SL C be a holomorphic map that can be continuously ex-tended to the circle A and such that F ( λ ) ∈ SU for all λ ∈ A . Then F holomorphicallyextends to A R into a map that satisfies t F (cid:18) λ (cid:19) = F ( λ ) − ∀ λ ∈ A R . (38) Proof.
Apply Schwarz reflexion principle on each coefficient of the matrix e F ( λ ) = (cid:18) F ( λ ) + F ( λ ) F ( λ ) − F ( λ ) i ( F ( λ ) + F ( λ )) i ( F ( λ ) − F ( λ )) (cid:19) where F ij denote the entries of F . The fact that F ( λ ) ∈ SU for all λ ∈ A ensures that Im e F = 0 on A . Thus, e F holomorphically extends to A R and satisfies for all λ ∈ A R e F (cid:18) λ (cid:19) = e F ( λ ) . Hence, F holomorphically extends to A R and satisfies F (cid:18) λ (cid:19) = F ( λ ) , F (cid:18) λ (cid:19) = − F ( λ ) which implies Equation (38) because F ( λ ) ∈ SL C . Corollary 3.
Let
Φ : A R −→ SL C be a holomorphic map and let F B be the Iwasawadecomposition of its restriction to A . Then F holomorphically extends to A R , satisfiesEquation (38) , and B holomorphically extends to D R .Proof. Write F = Φ B − to holomorphically extend F to A R , . Apply Lemma 7 toholomorphically extend F to A R , and write B = F − Φ to holomorphically extend B to D R . 28 Derivative of the monodromy
The following proposition, used in Section 3, is derived from Proposition 8 in [22].
Proposition 5.
Let ξ t be a C family of matrix-valued -forms on a Riemann surface Σ , defined for t in a neighbourhood of t ∈ R . Let e Σ be the universal cover of Σ . Fix apoint z in Σ and let e z be a lift of z to e Σ . Let Φ t be a continuous family of solutions of d Φ t = Φ t ξ t on e Σ such that for all t , (cid:2) M ( t ) , Φ t ( z )Φ t ( z ) − (cid:3) = 0 , where M ( t ) is the monodromy of Φ t with respect to some γ ∈ π (Σ , z ) . Let e γ be the liftof γ to e Σ such that e γ (0) = e z . Then M is differentiable at t and M ′ ( t ) = (cid:18)Z γ Φ t ∂ξ t ∂t | t = t Φ − t (cid:19) × M ( t ) . In particular, if M ( t ) = ± I or if Φ t ( z ) is constant, then (5) is satisfied.Proof. Proposition 8 in [22] is proved in the case where Φ t ( z ) is constant. Let e Φ t ( z ) =Φ t ( z ) − Φ t ( z ) , so that d e Φ t = e Φ t ξ t and e Φ t ( z ) = I n . Let f M ( t ) be the monodromy of e Φ t along γ . Then Proposition 5 of [22] applies and f M ′ ( t ) = (cid:18)Z γ e Φ t ( z ) ∂ξ t ( z ) ∂t | t = t e Φ t ( z ) − (cid:19) × f M ( t ) . On the other hand, M ( t ) = Φ t ( z ) f M ( t )Φ t ( z ) − and because of Equation (5), M ( t ) = Φ t ( z ) f M ( t )Φ t ( z ) − . Thus, M is differentiable at t and M ′ ( t ) = Φ t ( z ) f M ′ ( t )Φ t ( z ) − which proves the proposition. 29 A control formula on the unitary frame
The following proposition is used in Section 4.
Proposition 6.
Let (Σ , ξ, z , Φ z ) be a set of untwisted DPW data, holomorphic for λ ∈A R with R ≥ . Then for all z , z ∈ Σ and γ ⊂ Σ joining z to z , k F ( z , λ ) k A R ≤ C k F ( z , λ ) k A R × exp (cid:18) ( R − Z γ ρ ( w ) | a − ( w ) || dw | (cid:19) where C is a uniform positive constant, a − ( z ) dz is the λ − factor of ξ and ρ ( z ) is theupper-left entry of Pos(Φ)( z, .Proof. Write ξ ( z, λ ) = λ − (cid:18) a − ( z )0 0 (cid:19) dz + λ (cid:18) c ( z ) a ( z ) b ( z ) − c ( z ) (cid:19) dz + O ( λ ) . Let
Φ =
F B be the Iwasawa decomposition of Φ . Untwisting formula (4.3.5) of [5] withthe help of Remark 4.2.6 of [5] gives dF = F L where L ( z, λ ) = (cid:18) ρ − ρ z λ − ρ a − b ρ − − ρ − ρ z (cid:19) dz + (cid:18) − ρ − ρ ¯ z − b ρ − − λρ a − ρ − ρ ¯ z (cid:19) d ¯ z. Let e F ( z, λ ) = F (cid:18) z, λ | λ | (cid:19) so that e F ( z, λ ) ∈ SU for all λ ∈ A R . Then d e F = e F e L where e L ( z, λ ) = L (cid:18) z, λ | λ | (cid:19) . Using the variation of constants method, for all z , z ∈ Σ (we ommit the variable λ ), F ( z ) = F ( z ) e F ( z ) − e F ( z ) + (cid:18)Z z z F ( w ) (cid:16) L ( w ) − e L ( w ) (cid:17) e F ( w ) − (cid:19) e F ( z ) . But L ( w, λ ) − e L ( w, λ ) = ρ ( w ) (cid:18) a − ( w ) λ − (1 − | λ | ) dw − a − ( w ) λ (1 − | λ | − ) d ¯ w (cid:19)
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DPW method , (in preparation).Thomas RaujouanInstitut Denis PoissonUniversité de Tours, 37200 Tours, France [email protected]@univ-tours.fr