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Construction of constant mean curvature n-noids using the DPW method

  • Martin Traizet EMAIL logo

Abstract

We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.

A Derivative of the monodromy

The following proposition is adapted from [23, Proposition 9].

Proposition 8.

Let ξt be a C1 family of matrix-valued 1-forms on a Riemann surface Σ. Let Σ~ be the universal cover of Σ. Fix a point z0 in Σ and let z~0 be a lift of z0 to Σ~. Let Φt be a family of solutions of dΦt=Φtξt on Σ~ such that Φt(z~0) does not depend on t. Let γπ1(Σ,z0) and let M(t) be the monodromy of Φt with respect to γ. Let γ~ be the lift of γ to Σ~ such that γ~(0)=z~0. Then for all t,

M(t)=γ~ΦtξttΦt-1×M(t).

Proof.

Since Φt(z~0) is constant and ξt depends C1 on t, Φt depends C1 on t. Let Ψt=Φtt. By differentiation of the Cauchy Problem satisfied by Φt with respect to t, we obtain that Ψt satisfies the following Cauchy Problem on Σ~:

{dΨt=Ψtξt+Φtξtt,Ψt(z~0)=0.

Following the method of variation of constants , the function Ut=ΨtΦt-1 satisfies

{dUt=ΦtξttΦt-1,Ut(z~0)=0.

Hence (writing γ~(1) for the endpoint of γ~)

Ut(γ~(1))=γ~ΦtξttΦt-1.

We have by definition

M(t)=Φt(γ~(1))Φt(z~0)-1.

Hence since Φt(z~0) is constant,

M(t)=Ψt(γ~(1))Φt(z~0)-1
=Ut(γ~(1))Φt(γ~(1))Φt(z~0)-1
=γΦtξttΦt-1M(t).

B Smoothness of maps between Banach spaces

The following proposition is useful to prove that the maps considered in this paper are smooth maps between Banach spaces. The Banach algebra 𝒲ρ is defined in Section 4. For R>1, we denote by 𝔸R the annulus 1R<|λ|<R in . For 𝐚=(a1,,an)n and 𝐫=(r1,,rn)(0,)n, we denote by D(𝐚,𝐫) the polydisk i=1nD(ai,ri) in n.

Proposition 9.

Let R>ρ and let f:AR×D(a,r)C be a holomorphic function of (n+1) variables (λ,z1,,zn). Let

B(𝐚,𝐫)={(u1,,un)𝒲ρn:ui-ai<ri for all i[1,n]},

where we identify ai with a constant function in Wρ. Define for (u1,,un)B(a,r),

F(u1,,un)(λ)=f(λ,u1(λ),,un(λ)).

Then F:B(a,r)WρnWρ is of class C.

Proof.

We expand f in Laurent series with respect to λ and power series with respect to z1,,zn:

f(λ,z1,,zn)=ki1,,incki1inλk(z1-a1)i1(zn-an)in.

For any 𝐫<𝐫 (in the sense ri<ri for all i), the series f(ρ±1,a1+r1,,an+rn) converges absolutely so

(B.1)ki1,,in|cki1in|ρ|k|(r1)i1(rn)in<.

Let v𝒲ρ be the function defined by v(λ)=λ. Then formally,

(B.2)F(u1,,un)=ki1,,incki1invk(u1-a1)i1(un-an)in.

Since vk=ρ|k| and 𝒲ρ is a Banach algebra, for (u1,,un)B(𝐚,𝐫), we have

cki1invk(u1-a1)i1(un-an)in|cki1in|ρ|k|(r1)i1(rn)in

Hence by inequality (B.1), series (B.2) converges normally, so F(u1,,un)𝒲ρ and F is of class C on B(𝐚,𝐫). (See [1, Theorem 11.12] for the smoothness of maps defined by power series in Banach spaces.) ∎

Acknowledgements

I would like to thank the referee for several interesting suggestions (see Remark 4 and Section 10).

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Received: 2018-07-13
Revised: 2018-10-24
Published Online: 2018-12-18
Published in Print: 2020-06-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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