Finiteness of the set of solutions of Plateau's problem for polygonal boundary curves

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Finiteness of the set of solutions of Plateau’s problem for polygonal boundary curves

Ruben Jakob

ETHZ, Rämistr. 101, CH-8092 Zürich, Switzerland Received 1 September 2006; accepted 10 October 2006

Available online 26 February 2007

Abstract

It is proved that for a simple, closed, extremepolygonΓ ⊂R³every immersed, stable minimal surface spanningΓ is an isolated point of the set of all minimal surfaces spanningΓ w.r.t. theC⁰-topology. Since the subset of immersed, stable minimal surfaces spanningΓ is shown to be closed in the compact set of all minimal surfaces spanningΓ, this proves in particular thatΓ can bound onlyfinitelymany immersed, stable minimal surfaces.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:49Q05

Keywords:Finiteness of the number of minimal surfaces; Plateau’s problem for polygonal boundary curves

1. Introduction and main results

In 1978 Nitsche formulated the following conjecture (see [23]):A“reasonably well behaved”simple,closed con- tour can bound only finitely many solutions of Plateau’s problem. The aim of the present article is a proof of the following partial result:

Theorem 1.LetΓ ⊂R³be a simple, closed, extreme polygon. Then every immersed, stable minimal surface spanning Γ is an isolated point of the set of all minimal surfaces spanningΓ w.r.t. theC⁰-topology. In particular,Γ can bound onlyfinitelymany immersed, stable minimal surfaces.

A polygon is termedextremeif it is contained in the boundary of a compact, convex subset ofR³. Furthermore a disc-type minimal surfaceXis calledimmersedif there holds infB|DX|>0, where we denote byB the open unit disc{w=(u, v)∈R²| |w|<1}. It is said to bestableif the second variation of the area functionalAinXin normal directionξ:=(Xu∧Xv)/|Xu∧Xv|

* Tel.: 0041 44 63 23367.

E-mail address:[email protected] (R. Jakob).

1 Present address: Limmatstr. 29, CH-8005 Zürich, Switzerland.

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2006.10.003

J^X(ϕ):=

B

|∇ϕ|²+2KEϕ²dw= d²

dε²A(X+εϕξ )|ε=0 (1)

satisfiesJ^X(ϕ)0 for anyϕ∈C^∞_c (B), whereEdenotes|Xu|²andK0 the Gauss curvature ofX.

LetΓ be a closed Jordan curve inR³, and denote byC^∗(Γ )the Plateau classof surfaces X∈H^1,2(B,R³)∩ C⁰(B,R³)that spanΓ satisfying a fixed three-point condition. We endowC^∗(Γ )with the norm · _C⁰_(B)and denote by M(Γ ) its subspace {X∈C^∗(Γ )|ΔX=0, |Xu| = |Xv|, Xu, Xv =0 onB}of disc-type minimal surfaces.

Furthermore letMs(Γ )be the subspace ofM(Γ )consisting of those elements which are immersed and stable.

The first deep “finiteness result” was achieved by Tomi (1973) in [27]: IfΓ is a regular Jordan curve of classC^4,α with the property that all minimal surfacesX∈M(Γ ), which yield global minimizers of the area functionalAon C^∗(Γ ), are immersed, then there are only finitely many of them. In combination with the papers [7] resp. [15] Tomi’s theorem guarantees that regular Jordan curvesΓ which are either of classC^4,α with total curvature less than 4π or analytic can bound only finitely many global minimizers of the area functionalAonC^∗(Γ ). Four years later Tomi proved in [28]: IfΓ is a regular Jordan curve of classC^4,α, which bounds only minimal surfaces without boundary branch points and with interior branch points of at most first order (see Def. 1 below), thenMs(Γ )is finite. One year later Nitsche finally achieved his “6π-Theorem” in [23]: IfΓ is a regular Jordan curve of classC^4,α, which bounds only minimal surfaces without any branch points and whose total curvature does not exceed the value 6π, then the entiresetM(Γ )is finite.

In 1990, Sauvigny [24] proved a finiteness result for “small” H-surfaces which is very similar to Theorem 1. IfΓ is an extreme, regular Jordan curve of classC^4,α contained inB₁³(0)andH ∈ [0,1), then it can bound only finitely many immersed small H-surfacesXwhich are stable in the sense thatJ^X(ϕ)−4

BH²Eϕ²dw0∀ϕ∈C^∞_c (B).² The proofs of the above quoted results depend on boundary regularity results for minimal surfaces resp. H-surfaces spanningC^4,α-boundary curves due to Hildebrandt [16] resp. Heinz [8]. Analogs of these results are not available for polygons. Thus the author had to follow a completely different approach to prove Theorem 1 which uses fundamental results of Courant [1] and Heinz [10–13] in combination with ideas of Tomi [28] and Sauvigny [24–26]. Of particular importance are the asymptotic expansions for minimal surfaces in the corners of the bounding polygon due to Heinz [11] and Heinz’ discovery of a deep connection between the total branch point orders of such minimal surfaces, the defects of their assigned Schwarz operators and the number of vertices of the bounding polygon, expressed in his formula (12) below, the main result of his paper [14].

2. Courant’s and Heinz’ results

A polygonΓ is a closed piecewise linear Jordan curve inR³withN+3 vertices (N∈N)(P₁, . . . , P_N+3), where we require the pairs of vectors (Pj+1−P_j,P_j−P_j−1) to be linear independent forj =1, . . . , N+3, withP₀:=P_N+3 andPN+4:=P1. We consider the “Plateau class”C^∗(Γ )of surfacesX∈H^1,2(B,R³)∩C⁰(B,R³)that are spanned intoΓ, i.e. whose boundary valuesX|∂B:S¹→Γ are weakly monotonic mappings with degree equal to 1, satisfying a three-point condition:X(e^iτN+k)=PN+k forτN+k:=^π₂(1+k),k=1,2,3. Our fundamental tools are Courant’s [1] and Heinz’ [10], [11] maps

ψ:T −→

C^∗(Γ ), · _C⁰_(B)

, ψ˜:T −→C⁰ B,R³

∩C² B,R³

, (2)

which are assigned to our arbitrarily fixed closed polygonΓ. HereT is an open, bounded, convex set ofN-tuples (τ₁, τ₂, . . . , τ_N)=:τ ∈(0, π )^N, which meet the following chain of inequalities 0< τ₁<· · ·< τ_N< π=τ_N+1, where N+3 was the number of vertices of the considered polygon. Moreover to anyτ∈T we assign the sets of surfaces

U(τ ):=

X∈C^∗(Γ )|X|∂B(e^iτj)=P_jforj =1, . . . , N and U(τ ):=

X∈C⁰ B,R³

∩C² B,R³

|X e^iθ

∈Γj forθ∈ [τj, τj+1] ,

2 After completion of this manuscript, Prof. Frank Morgan kindly communicated one of his results to the author [see Indiana Univ. Math. J. 35 (4) (1986) 813, Theorem 5.8] which generalizes Sauvigny’s theorem especially to systems ofC^2,α-Jordan curves lying on the boundary of an arbitrary strictly convex set inR³, but only forH=0. The smoothness of the boundary curves are crucial in his proof, as in the results of Tomi, Nitsche, and Sauvigny.

for 1jN+3, where we setΓj:= {Pj+t (Pj+1−Pj)|t∈R},PN+4:=P1andτN+4:=τ1. On account of two uniqueness results in [1] resp. [10] one can define the maps

ψ (τ ):=unique minimizer ofDwithinU(τ ) and ψ(τ )˜ :=unique minimizer ofDwithinU(τ ),

whereDdenotes Dirichlet’s integral. We will also use the notationX(·, τ )forψ(τ ). Now by the result of [1] (see also˜ [18], p. 558) Satz 1 and 2 in [10] and the main theorem of [11], resp. Satz 1 in [13], these maps have the following properties:

Theorem 2.

(i) ψis continuous onT.

(ii) f :=D◦ψis of classC¹(T )andf˜:=D◦ ˜ψeven of classC^ω(T ).

(iii) There holds f˜f on T and f (τ )˜ =f (τ ) if and only if ψ(τ )˜ =ψ (τ ), which is again equivalent to ψ (τ )˜ ∈C^∗(Γ ).

(iv) ψ(τ )˜ andψ (τ )are harmonic onB ∀τ∈T. (v) The restriction

ψ|K(f ):K(f )−→^∼= M(Γ ) (3)

yields ahomeomorphismbetween the compact set of critical points off and(M(Γ ), · _C⁰_(B))and a surface ψ (τ )˜ is conformally parametrized onB, thus a minimal surface inU(τ ), if and only ifτ∈K(f ).˜

(vi) Letτ¯∈T be arbitrarily fixed andD⊂Csome simply connected domain whose intersectionD∩B withB is nonvoid and connected and such thatD¯∩ {e^iτ¯k} = ∅. Then there exists some neighborhoodUD(τ )¯ ofτ¯inC^N and some holomorphic continuation ofXw(·,·)ontoD×UD(τ ).¯

(vii) Furthermore for anyτ¯∈T andk∈ {1, . . . , N+3}there exists some neighborhoodBδ(e^iτ¯k)×B_δ^N(τ )¯ inC×C^N about(e^iτ¯k,τ )¯ such that there holds the representation

X_w(w, τ )=

pk

j=1

f_j^k(w, τ )

w−e^iτkρ_j^k

(4) for(w, τ )∈(Bδ(e^iτ¯k)∩B)×(B_δ^N(τ )¯ ∩R^N), where the functionsf_j^kare holomorphic onBδ(e^iτ¯k)×B_δ^N(τ )¯ and the exponentsρ_j^ksatisfy

−1< ρ₁^k<· · ·< ρ_p^k

k=0, p_k∈ {2,3}, (5)

and do not depend onτ∈B_δ^N(τ )¯ ∩R^N.

The last assertion about the independence of the exponentsρ_j^kofτ ∈B_δ^N(τ )¯ ∩R^Nfollows immediately from [10], (2.20) and (3.28), as we point out now. We setv_k:=(P_k+1−P_k)/|P_k+1−P_k|and consider as in (2.20) of [10] the reflectionsS_k at the linesΓ_k−P_k=Span(vk)fork∈ {1, . . . , N+3}(withP_N+4:=P₁), explicitly given by

S_k(x):= −x+2v_k, xv_k ∀x∈R³.

The composed reflectionsS_k−1◦S_k∈SO(3)are diagonalizable by conjugation with unitary matrices and have eigen- values on theS¹. Now theρ_j^k appear in (3.28) of [10] as pairwise different (negative) angles of these eigenvalues, precisely:

Spec(Sk−1◦S_k)=

e^−2πiρkj

, 1jp_k,

ordered as in (5) withp_k∈ {2,3}, which proves the claimed independence of the exponentsρ_j^k ofτ ∈B_δ^N(τ )¯ ∩R^N. Moreover we shall note that S_k−1◦S_k =1 and thus p_k >1 by our requirement that the vectorsP_k−1−P_k and P_k+1−P_khave to be linearly independent. Moreover we see thatp_k=2 if and only ifρ₁^k= −¹₂, i.e. if the spectrum of S_k−1◦S_k is {−1,−1,1}, which can arise if and only if the smaller angleβ_k between the vectors P_k−1−P_k andP_k+1−P_k is ^π₂. If in generalβ_k ∈ {/ ^π₂,0, π}, then the spectrum of S_k−1◦S_k is{λ,λ,¯ 1}for someλ∈S¹with

(λ)=0, i.e.ρ^k₁+ρ₂^k= −1. One can easily see that there holds either−ρ^k₁π=βk or(ρ₁^k+1)π=βk, which is by ρ₁^k+ρ₂^k= −1 equivalent to the pair of possibilities(ρ₂^k+1)π=β_k or−ρ₂^kπ =β_k. Moreover we can expand the holomorphic functionsf_j^kw.r.t.wabout the point e^iτk and obtain by (4) for anyk∈ {1, . . . , N+3}:

X_w(w, τ )=

pk

j=1

∞ n=0

f_j,n^k (τ )

w−e^iτkρ_j^k+n

(6)

∀w∈B_δ(e^iτ¯k)∩B and∀τ∈B_δ^N(τ )¯ ∩R^N. Now we fix someτ¯∈T and choose that pair(j, n)for whichf_j,n^k (τ )¯ =0 in (6) andρ_j^k+nis minimal and term this pair(j^∗, m), i.e. we assign this pair to the pointτ¯∈T. Since we know that either(ρ_j^k_∗+1)π or−ρ_j^k_∗π equals the smaller angleβ_k=0, π between the linear independent vectorsP_k−1−P_k andPk+1−Pkwe conclude due toρ_p^k

k=0 that there has to holdj^∗< pk. Now using these terms the author derived in [20], Corollary 2.1:

Corollary 1.For any fixedτ¯∈T andk∈ {1, . . . , N+3}there holds X_w(w,τ )¯ =f_j^k_∗,m(τ )¯

w−e^iτ¯kρ_j∗^k +m

+Ow−e^{iτ¯kρkj∗+m+ε}

(7) forBw→e^iτ¯k, whereε:=ρ^k_j∗+1−ρ_j^k_∗∈(0,1).

Furthermore we derive from part (vi) of Theorem 2 that there is a Taylor expansion ofX_w(·,τ )¯ about any point w₀∈B\ {e^iτ¯k}:

X_w(w,τ )¯ =a_m(τ )(w¯ −w0)^m+a_m+1(τ )(w¯ −w0)^m+1+ · · ·, (8) where the coefficients{a_j}jmare holomorphic about the pointτ¯anda_m(τ )¯ ∈C³\ {0}.

Definition 1.

(i) We term the exponentm≡m^τ¯ in (7) resp. (8) the branch point order of the surfaceX(·,τ )¯ at the point e^iτ¯k, k=1, . . . , N+3, resp.w₀∈B\ {e^iτ¯k}.

(ii) A pointw∈Bis termed a branch point of the minimal surfaceX(·,τ )¯ ∈M(Γ )if its orderm^τ¯(w)is positive.

Hence, we see that there holdsm^τ¯(w)=0 in any pointw∈Bif and only if infB

DX(·,τ )¯ >0. (9) Furthermore one obtains easily by (7) and (8) thatX(·,τ )¯ can have only finitely many branch points onB. Hence, we may define its total branch point order

κ(τ )¯ :=

w∈B

m^τ¯(w)+1 2w∈∂B

m^τ¯(w). (10)

Now we assign to every pointτ∈K(f ), i.e. to every minimal surface˜ X(·, τ ), its Schwarz operator A^τ≡A^{X(·,τ )}:= −Δ+2(KE)^τ,

where(KE)^τ(w):=(KE)(w, τ )is defined as in (1), on the domain Dom(A^τ):=

ϕ∈C²(B)∩H˚^1,2(B)|A^τ(ϕ)∈L²(B)

(11) and formulate as a central tool of our paper the “Heinz’ formula” from [14]: For an arbitraryτ ∈K(f )˜ one has

dim KerA^τ+rank

D²f (τ )˜

+2κ(τ )=N. (12)

Furthermore by differentiation of (6) w.r.t.w we obtain exactly as in the proof of Corollary 1 on account of the holomorphy of the components ofX_w(·,τ )¯ onB:

X_ww(w,τ )¯ =f_j^k_∗,m(τ )¯

m+ρ^k_j∗

w−e^iτ¯kρ^k_j∗+m−1

+Ow−e^{iτ¯kρkj∗+m+ε−1}

, (13)

forBw→e^iτ¯k. Furthermore (cf. formula (2.8) in [14]) one proves by integration ofXw(·,τ )¯ in (6) w.r.t.wthat for any fixedτ¯∈T andk∈ {1, . . . , N+3}there exists someδ >0 such that

X(w, τ )=2 w

eⁱτk

X_z(z, τ )dz

+X e^iτk, τ

= _pk

j=1

g^k_j(w, τ )

w−e^iτkρ_j^k+1

+P_k (14)

forw∈Bδ(e^iτ¯k)∩Bandτ ∈B_δ^N(τ )¯ ∩R^N, where the functions g_j^k(w, τ ):=

∞ n=0

2f_j,n^k (τ ) ρ_j^k+n+1

w−e^iτkn

are holomorphic onB_δ(e^iτ¯k)×B_δ^N(τ )¯ and satisfy in particularg^k_j(e^iτk, τ )=2f_j,0^k (τ )/ρ_j^k+1. Next, as stated in for- mula (2.9) in [14], one achieves by differentiation of (14) w.r.t.τ_lforl∈ {1, . . . ,k, . . . , Nˆ }:

Xτ_l(w, τ )= _p

k

j=1

∂g_j^k

∂τ_l(w, τ )

w−e^iτkρ_j^k+1

, (15)

forw∈B_δ(e^iτ¯k)∩Bandτ ∈B_δ^N(τ )¯ ∩R^Nand forl=k:

X_τl(w, τ )= _pl

j=1

∂g_j^l

∂τ_l(w, τ )

w−e^iτlρ_j^l+1

−ie^iτl ρ_j^l +1

g_j^l(w, τ )

w−e^iτlρ_j^l

. (16)

To verify now formula (2.10) in [14] we set _∂ϕ^∂ :=u_∂v^∂ −v_∂u^∂ and compute for some l =k∈ {1, . . . , N} and j∈ {1, . . . , pl}

∂

∂ϕ

w−e^iτlρ_j^l+1

=

ρ_j^l +1

w−e^iτlρ_j^l

iw, (17)

thus obtaining forl=k:

X_τl(w, τ )+X_ϕ(w, τ )= _pl

j=1

∂g^l_j

∂τl

(w, τ )+∂g^l_j

∂ϕ (w, τ )+i ρ_j^l +1

g^l_j(w, τ )

w−e^iτlρ_j^l+1

(18)

forw∈Bδ(e^iτ¯l)∩Bandτ∈B_δ^N(τ )¯ ∩R^N. Combining this with (15) we achieve

Corollary 2.Letτ¯∈K(f )˜ be arbitrarily chosen. If there holdsm^τ¯(e^iτ¯l)=0for eachl∈ {1, . . . , N}, then the functions X_τl(·,τ )¯ are linearly independent onB.

Proof. Otherwise there would exist some linear relation

N

l=1

α_lX_τl(·,τ )¯ ≡0 onB, (19)

where there is at least one indexk∈ {1, . . . , N}withα_k=0. By (15) we see thatX_τl(w,τ )¯ →0 forw→e^iτ¯k and l=k. Hence, inserting this into (19) we obtain due toα_k=0 thatX_τk(w,τ )¯ →0 forw→e^iτ¯k. Now together with (18) we conclude that there holds alsoX_ϕ(w,τ )¯ →0 forw→e^iτ¯k. Moreover as we requireτ¯∈K(f ), thus that˜ X(·,τ )¯ has to be conformally parametrized onB, we have|X_ϕ(w,τ )¯ |²=2|w|²|X_w(w,τ )¯ |²,∀w∈B. Hence, we would finally obtain X_w(w,τ )¯ →0 forw→e^iτ¯k, which would implym^τ¯(e^iτ¯k) >0 by (7), contradicting the requirement of the corollary. 2

Now we set ρ:= min

k=1,...,N+3ρ₁^k>−1. (20)

By the Courant–Lebesgue Lemma and point (iv) of Theorem 2 the author proved in Chapter 2 in [20] the following important

Lemma 1.There holds

M(Γ ):= {set of minimal surfaces onB} ∩

τ∈T

U(τ )∩H^1,2 B,R³

=

X∈image(ψ )˜ |Xis also conformally parametrized onB .

Combining this result with (8) and point (v) of Theorem 2 the author proved in Chapter 2 in [20]

Corollary 3.There holdsM(Γ )⊂M(Γ )and alsoK(f )⊂K(f ). In particular˜ X(·, τ )≡ ˜ψ(τ )coincides withψ (τ ) for anyτ ∈K(f ).

Moreover let

ξ(·, τ ):= X_u∧X_v

|X_u∧X_v|(·, τ )=X_w∧X_w

i|X_w|² (·, τ ) (21)

denote the unit normal field of some minimal surfaceX(·, τ )∈M(Γ ), i.e. for someτ ∈K(f ). By (7) and (8) one˜ achieves thatξ(·, τ )can be continued continuously ontoBand even analytically ontoB\ {e^iτl}l=1,...,N+3, although it is not defined in the branch points ofX(·, τ ), and that at some point e^iτk it behaves asymptotically like

ξ(w, τ )=f_j^k_∗,m(τ )∧f_j^k_∗,m(τ )

i|f_j^k_∗,m(τ )|² +Ow−e^iτkε

forw−→e^iτk (22)

andk=1, . . . , N+3. Together with (13) one obtains moreover:

|ξ(w, τ ), X_ww(w, τ )|

|X_w(w, τ )| =Ow−e^iτkε−1

forw−→e^iτk (23)

andk=1, . . . , N+3. Heinz used this and identity (3.2) in [14], (KE)^τ(w)≡(KE)(τ, w)= −8|ξ(w, τ ), Xww(w, τ )|²

|X_w(w, τ )|² (24)

for anyτ ∈K(f ), in order to prove: There is some constant˜ const.(τ ), depending onτ andΓ only, such that:

(KE)^τ(w)const.(τ )

N+3

k=1

w−e^iτk−2+α ∀w∈B, (25)

for

α:=2 min

ρ_j^k−ρ_j^k₋₁|j=1, . . . , pk, k=1, . . . , N+3

>0 (26)

where we setρ₀^k := −1 for eachk. Thus α only depends onΓ but not on τ ∈K(f ), as the˜ ρ_j^k do not, whence (KE)^τ ∈L^p∗(B)for anyp^∗∈(1,₂₋²_α)and anyτ∈K(f ). Moreover one can insure by (24) and (8) that the function˜ (KE)^τ, which is not defined in the branch points ofX(·, τ ), is of classL^∞_loc(B\ {e^iτl}l=1,...,N+3)and can in fact be continued analytically ontoB\ {e^iτl}l=1,...,N+3.

3. Compactness ofMs(Γ )

Theorem 3.IfΓ is a rectifiable, closed Jordan curve inR³which bounds only minimal surfaces without boundary branch points, thenMs(Γ )is a closed subset ofM(Γ ), thus compact, w.r.t. theC⁰(B)-topology.

Proof. Let{Xⁿ}be some sequence inMs(Γ )converging to someX∈M(Γ )inC⁰(B,R³). Then we inferXⁿ→X inC_loc¹ (B,R³)from Cauchy’s estimates and thus thatX also cannot have any interior branch points by Theorem 1 in [24], where one has to use that X cannot be a constant map on account of the imposed three-point condition.

Moreover, sinceXis bounded byΓ it must be free of boundary branch points, and thus immersed onB. Secondly we show the stability ofX. To this end we fix someϕ∈C_c^∞(B)arbitrarily. On account of the requirement|Xⁿ_w|>0 on Bwe can use the identity(KE)ⁿ= −8|ξⁿ, Xⁿ_ww|²/|Xⁿ_w|²in order to conclude by Cauchy’s estimates that

(KE)ⁿϕ²(w)−→KEϕ²(w) pointwise for a.e.w∈B

and forn→ ∞. Now together with−(KE)ⁿ0 onB andJ^Xn(ϕ)0 for anyn∈Nwe achieve by Fatou’s lemma:

B

−2KEϕ²dwlim inf

n→∞

B

−2(KE)ⁿϕ²dw

B

|∇ϕ|²dw,

i.e.J^X(ϕ)0 for anyϕ∈C_c^∞(B). Hence,Ms(Γ )is a closed subset ofM(Γ )and thus compact on account of the well-known compactness ofM(Γ ). 2

Now we fix again some simple, closed polygonΓ and prove the following approximation result to be used below in Sections 6 and 7:

Lemma 2. If X is some stable minimal surface in M(Γ ), then there holds J^X(ϕ)0 even for all functions ϕ∈H˚^1,2(B).

Proof. By Lemma 1 and point (v) of Theorem 2 we know that there exists someτ ∈K(f )˜ such thatX= ˜ψ (τ ). Thus we conclude by (26) that there exists somep^∗>1 such thatKE∈L^p∗(B). Now letϕ∈H˚^1,2(B)be chosen arbitrarily and{ϕ_j} ⊂C_c^∞(B)some sequence withϕ_j→ϕ in ˚H^1,2(B). By Sobolev’s embedding theorem we haveϕ_j→ϕin L^q(B), for anyq∈ [1,∞), and therefore together with Hölder’s inequality: KE(ϕ_j²−ϕ²) _L1(B)→0, forj → ∞.

Thus together with the required stability ofXwe obtain 0J^X(ϕ_j)→J^X(ϕ)forj→ ∞, henceJ^X(ϕ)0. 2 4. Extreme polygons prevent boundary branch points

In this section we collect the author’s results of Chapter 4 of [20] which guarantee that Theorem 3 especially applies toextremepolygons. In Chapter 4 of [20] the author proved the following generalization of Hopf’s “boundary point lemma”:

Lemma 3.LetΦ∈C⁰(B)∩C²(B)be harmonic onBand satisfy

Φ(w0) > Φ(w) ∀w∈B, (27)

for some fixed pointw₀∈∂B. Then there exists some constantσ >0such that there holds Φ(w₀)−Φ(w)

|w₀−w| > σ ∀w∈K_δ,^π

4(w₀), (28)

for some sufficiently small chosenδ >0, whereK_δ,^π

4(w0):= {w∈Bδ(w0)∩B||angle(w−w0,−w0)| ∈ [0,^π₄]}.

By this result the author derived in Chapter 4 of [20]:

Theorem 4.If Γ is an extreme, simple, closed polygon which is not contained in a plane, then a minimal surface X∈M(Γ )does not possess any boundary branch points.

Secondly, for an arbitrary simple, closed polygonΓ one has

Lemma 4. Let X(·, τ )∈M(Γ ) be a minimal surface whose boundary values are not monotonic on some arc (e^iτk,e^iτk+1)⊂S¹, for somek=1, . . . , N+3. Then there exists some angleθ∈(τk, τk+1)for whiche^iθis a boundary branch point ofX(·, τ ). In particular, anyX∈M(Γ )\M(Γ )possesses a boundary branch point.

Thirdly, a combination of this lemma with the proof of Theorem 4 implies

Corollary 4.LetΓ be some extreme, simple, closed polygon. If a minimal surfaceX∈M(Γ ) satisfies2κ(X)=1 then it is already free of branch points onB.

Combining (24) with Theorem 1 on p. 175 in [3] the author derived in Chapter 4 of [20] for any extreme, simple, closed polygonΓ:

Corollary 5. Let τ¯ ∈K(f ) and δ >0 be fixed with the property that X(·, τ ) has no branch points on B for τ∈Bδ(τ )¯ ∩K(f ), where we set Bδ(τ )¯ :=B_δ^N(τ )¯ ∩R^N. Then there exists someδ¯∈(0, δ] and some constant C depending onΓ,τ¯andδ¯only such that there holds

(KE)^τ(w)C

N+3

k=1

w−e^iτk−2+α ∀w∈B, (29)

for anyτ∈K(f )∩B_δ¯(τ ), with¯ α:=2 min{ρ_j^k−ρ_j^k₋₁|j=1, . . . , pk, k=1, . . . , N+3}>0andρ₀^k:= −1for eachk.

5. The Schwarz operatorsA^τ andA˙^τ forτ∈K(f )˜ We set

C₀²(B):=

ϕ∈C²(B)∩C⁰(B)|ϕ|∂B≡0

and consider the Schwarz operatorA^τ:= −Δ+2(KE)^τ, forτ∈K(f ), on the domain˜ Dom(A^τ):=

ϕ∈C²(B)∩H˚^1,2(B)|A^τ(ϕ)∈L²(B)

, (30)

and the minimal Schwarz operatorA˙^τ and minimal Laplace operatorΔ˙ on the domain H^2,2(B)∩C₀²(B). Using estimate (25) one can prove assertion (3.11) in [14] (see Proposition 2.1 in [19] or Section 7.1 in [20]) which reads:

Proposition 1.For anyϕ∈H^2,2(B)∩C₀²(B)and anyτ∈K(f )˜ there holds (KE)^τϕ(w)c(τ, α)

N+3

k=1

w−e^{iτk−1+α/2} ˙Δϕ _L2(B) ∀w∈B. (31)

Now in Proposition 2.2 in [19] the author proved thatH^2,2(B)∩C₀²(B)is densely contained inH^2,2(B)∩H˚^1,2(B) w.r.t. theH^2,2(B)-norm, which implies that estimate (31) extends ontoH^2,2(B)∩H˚^1,2(B)for anyτ ∈K(f ). Using˜ these results the author achieved in [19] (or Section 7.1 in [20]) that there holds for anyϕ∈H^2,2(B)∩H˚^1,2(B)and anyτ∈K(f ):˜

2(KE)^τϕ

L²(B)1

2 Δϕ _L2(B)+c ϕ _L2(B), (32)

for some constantc=c(τ ) that only depends onτ. Using this estimate the author proved in [19] that Dom(Δ)¯˙ = Dom(A˙^τ)=H^2,2(B)∩H˚^1,2(B),∀τ ∈K(f ). Moreover in [19] or Section 7.2 in [20] the author showed the essential˜ self-adjointness ofΔ˙w.r.t.·,·L²(B), i.e.Δ¯˙ =(Δ)¯˙ ^∗. Together with estimate (32), forτ ∈K(f ), one can infer from˜ Theorem 4.4 in [21], p. 288, that alsoA˙^τ = − ˙Δ+2(KE)^τ is essentially self-adjoint w.r.t.·,·L²(B),∀τ ∈K(f ).˜

Furthermore it is proved in [19] thatA^τ is symmetric, i.e.A^τ⊂(A^τ)^∗, and thus closable inL²(B). Hence, together with the fact that Dom(A^τ)is densely contained inL²(B)one can derive by twice application of Theorem 5.29 in [21], p. 168, that(A^τ)^∗is densely defined inL²(B)and closed,(A^τ)^∗∗= ¯A^τ and(A^τ)^∗=(A^τ)^∗=((A^τ)^∗)^∗∗,∀τ ∈K(f ).˜ Combining the above results with Theorem 5.29 in [21], p. 168, the author achieved in [19] or Section 7.2 in [20]:

Theorem 5.(A˙^τ)^∗= ˙A^τ= ¯A^τ=(A^τ)^∗are self-adjoint operators with domainH^2,2(B)∩H˚^1,2(B),∀τ∈K(f ).˜ Now we will denote SH˚^1,2(B):= {ϕ∈H˚^1,2(B)| ϕ _L2(B)=1}, and analogouslyS(H^2,2(B)∩H˚^1,2(B))and SDom(A^τ). Moreover we consider the bilinear form

L^τ(ϕ, ψ ):=

B

∇ϕ· ∇ψ+2(KE)^τϕψdw,

for ϕ, ψ∈H˚^1,2(B), and denoteJ^τ(ϕ):=L^τ(ϕ, ϕ). We fix someτ ∈K(f )˜ andp^∗∈(1,₂₋²_α)arbitrarily and ab- breviateA:=A^τ,L:=L^τ andJ :=J^τ. In Section 4 of [19] or Chapter 8 in [20] the author proved by Ehrling’s interpolation lemma that there exists some constantC(p^∗)such that

J (ϕ)1 2

B

|∇ϕ|²dw−C(p^∗) KE _L^p∗_(B) ∀ϕ∈SH˚^1,2(B). (33)

Combining this result with L²-regularity theory, Theorem 8.13 in [5], Theorem 5 and well known ideas of spectral theory the author achieved in [19] or Chapter 8 in [20]:

Theorem 6.The spectra ofAandA¯coincide, are discrete and accumulate only at∞, thus their eigenspaces are finite dimensional. Furthermore there holds for their common smallest eigenvalue:

λ_min(A)= inf

SDom(A)J= inf

SH˚^1,2(B)

J= inf

S(H^2,2(B)∩H˚^1,2(B))

J=λ_min(A).¯ (34)

Thus we may abbreviate λ_min:=λ_min(A)=λ_min(A). Moreover, applying Harnack’s inequality, Theorem 8.20¯ in [5], to the modulus |ϕ^∗|of some eigenfunctionϕ^∗∈ESλmin(A)¯ ⊂H^2,2(B)∩H˚^1,2(B), with ϕ^∗ _L2(B)=1, the author finally derived from the above theorem (see Theorem 1.2 in [19]):

Theorem 7.

(i) For an eigenfunctionϕ^∗∈ESλ_min(A)¯ there holds|ϕ^∗|>0onBand therefore:

dimESλmin(A)¯ =dimESλmin(A)=1. (35)

(ii) Especially an eigenfunctionϕ^∗∈ESλ_min(A)satisfies|ϕ^∗|>0onB. 6. The componentK(f )˜ ¹_τ∗ ofK(f )˜ is a closedC^ω-curve

From this section on we assume thatΓ is an extreme, simple, closed polygon which is not contained in a plane.

We shall prove the main result, Theorem 1, by contradiction. Thus we assume the existence of someX^∗∈Ms(Γ ) and some sequence{Xⁿ} ⊂M(Γ )with

Xⁿ−→X^∗ inC⁰ B,R³

. (36)

Thus by means of (3) the pointsτ^∗:=ψ⁻¹(X^∗)∈Ks(f )andτⁿ:=ψ⁻¹(Xⁿ)∈K(f )satisfy

τⁿ−→τ^∗ inK(f ), (37)

where we introduced the notation K_s(f ):=ψ⁻¹

Ms(Γ ) .

ByK(f )⊂K(f ) τ˜ ^∗would be a non-isolated critical point off˜and therefore rank(D²f (τ˜ ^∗))N−1. Moreover we know thatX(·, τ^∗)≡ ˜ψ(τ^∗)coincides withψ (τ^∗)=X^∗∈Ms(Γ )by Corollary 3. Hence, we haveκ(τ^∗)=0 and could conclude now by Heinz’ formula (12) dim KerA^X(·,τ∗)1, thus 0 would be an eigenvalue ofA^τ∗:=A^X(·,τ∗). Moreover we know together with Lemma 2 that there holds

J^τ∗:=J^X(·,τ∗)0 on ˚H^1,2(B), (38)

thus in particular on Dom(A^τ∗). Therefore 0 would even be the smallest eigenvalue ofA^τ∗. For if there were some negative eigenvalueλ^∗<0 ofA^τ∗with some eigenfunctionϕ^∗, we would obtain by the proof of the symmetry ofA^τ∗ in [19] that

J^τ∗(ϕ^∗)=L^τ∗(ϕ^∗, ϕ^∗)=

A^τ∗(ϕ^∗), ϕ^∗

L²(B)=λ^∗ϕ^∗, ϕ^∗L²(B)<0, (39)

which is a contradiction. Hence, together with (35) we would arrive at dim Ker

A^τ∗

≡dimESλmin=0

A^τ∗

=1. (40)

In combination withκ(τ^∗)=0 we could therefore derive from formula (12) exactly rank(D²f (τ˜ ^∗))=N−1. Hence, there would be aδ >0 such that

rank D²f˜

N−1 onBδ(τ^∗), (41)

where we abbreviateBδ(τ^∗)for B_δ^N(τ^∗)∩R^N⊂⊂T. Due to f˜∈C^ω(T ) we know thatK(f )˜ is an analytic set which possesses therefore a locally finite analytic triangulation due to [22], p. 463, and is therefore especially locally connected, such that a combination of (41) and (37) shows that the sequence {τⁿ} would approachτ^∗ along the 1-skeletonK(f )˜¹of the simplicial complexK(f ). Hence, we would arrive at the˜

Contradiction hypothesis.If the assertion of the main result, Theorem 1, were wrong, then there would have to exist some pointτ^∗∈K_s(f )which is also contained in the closure of the union of 1-simplices of the simplicial complex K(f ), i.e.˜ τ^∗∈K_s(f )∩(K(f )˜¹\K(f )˜⁰)= ∅.

Now this gives rise to the idea to analyze the following subsetZ of the connected componentK(f )˜ ¹_τ∗ of the 1-skeletonK(f )˜¹(ofK(f )) that contains˜ τ^∗:

Z:=

τ∈K(f )˜¹_τ∗|κ(τ )=0,rank

D²f (τ )˜

=N−1, J^τ 0 onC_c^∞(B) . Now we prove the following crucial

Theorem 8.The setZ(= ∅) is an open and closed subset ofK(f )˜¹_τ∗, thusZ=K(f )˜ ¹_τ∗. Proof. (a) Byτ^∗∈Zwe know thatZis not empty.

(b) Secondly we derive the “openness” of the conditionκ(τ )=0, i.e. we prove: Letτ¯∈Zbe some arbitrarily fixed point, then there exists someδ >0 such thatκ≡0 onB_δ(τ )¯ ∩K(f ). In fact, since we have rank˜ D²(f )(˜ τ )¯ =N−1 there exists someδ >0 such that rankD²(f )(τ )˜ N−1 for anyτ∈B_δ(τ ). Hence, together with Heinz’ formula (12)¯ we can conclude that 2κ(τ )1 for anyτ ∈B_δ(τ )¯ ∩K(f ). Thus recalling Corollary 4 we achieve in fact˜ κ(τ )=0 for anyτ ∈B_δ(τ )¯ ∩K(f ).˜

(c) Next we show thatκ(τ )=0 andJ^τ0 onC_c^∞(B)are “closed conditions”. To this end we combine the above reasoning with Corollary 4 in order to achieve: Letτ¯∈Z be some arbitrarily fixed point, then there exists some δ >0 such that there holdsK(f )˜ ∩B_δ(τ )¯ =K(f )∩B_δ(τ ). In particular, this shows¯ Z⊂K(f ). This can be seen as follows: The inclusion “⊃” follows from Corollary 3. “⊂”: By part (b) of the proof we know that there exists some δ >0 such that there holdsκ≡0 onK(f )˜ ∩B_δ(τ ). Now if the assertion were wrong there would have to exist some¯ pointτ^∗∈K(f )˜ ∩B_δ(τ )¯ which is contained inK(f )˜ \K(f )and thereforeX(·, τ^∗)∈M(Γ )\M(Γ )on account of Lemma 1 and points (v) and (iii) of Theorem 2. By Corollary 4 this implies thatX(·, τ^∗)would have to possess a boundary branch point in contradiction toκ(τ^∗)=0. Now the second assertion follows from the first one due to Z⊂K(f )˜ by its definition. Now we consider a sequence{τⁿ} ⊂Z that converges to some pointτˆ∈K(f )˜ ¹_τ∗. On