<span class="mathjax-formula">$H$</span>-surface index formula

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www.elsevier.com/locate/anihpc

H-surface index formula

Ruben Jakob

¹

Mathematisches Institut der Universität, Bonn, Germany Received 6 June 2004; accepted 6 October 2004

Available online 23 May 2005

Abstract

We construct an additive indexIon the set of compact “parts” of the setHH(Γ )of “small” H-surfaces (|H|<¹₂) that are spanned into a simple closed polygonΓ ⊂R³withN+3 vertices (N1) by a combination of Heinz’ and Hildebrandt’s examinations of H-surfaces and Dold’s fixed point theory. We obtain that the index ofHH(Γ )is always 1, independent ofH andΓ. Moreover we compute that the ˇCech cohomologyH (ˇ P)of a partPthat minimizes the H-surface functionalE^H locally is non-trivial at most in degrees 0, . . . , N−1 and there even finitely generated, which implies the finiteness of the number of connected components ofPin particular. Finally the index of such an “E^H-minimizing” part reveals to coincide with its Cech–Euler characteristic, which yields a variant of the mountain-pass-lemma.ˇ

1. Introduction and main result

LetΓ be some closed piecewise linear Jordan curveΓ ⊂D³:= {x∈R³| |x|1}withN+3 vertices (N∈N) (P₀, A₁, . . . , A_l;P₁;A_l₊₁, . . . , A_m;P₂;A_m₊₁, . . . , A_N), (1) where the three verticesP0,P1,P2 and the indices 0lmN are fixed. We consider the (Plateau-) class C₁^∗(Γ )of surfacesX∈H^1,2(B,R³)∩C⁰(B,R³),B:=B₁(0)⊂R², 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|∂B(e^iψ^k)=P_k, ψ_k:=k2π

3 , k=0,1,2, (2)

and that are contained in the closed unit ballD³, i.e.XC⁰(B)1. We endowC^∗₁(Γ )with the norm · H^1,2∩C⁰ :=

· _C0(B)+ · _H^1,2_(B). Moreover we consider the subspace HH(Γ )⊂(C₁^∗(Γ ), · _H^1,2_∩_C⁰) of all “small”

H-surfaces, i.e. classical solutionsX∈C₁^∗(Γ )∩C²(B,R³)of the differential equations

E-mail address: ruben@math.uni-bonn.de (R. Jakob).

1 It is a pleasure for the author to thank Prof. Dr. S. Hildebrandt for his support.

doi:10.1016/j.anihpc.2004.10.008

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X=2H (X_u∧X_v) onB, (3)

|X_u|²= |X_v|², X_u, X_v =0 onB, (4) for some arbitrarily fixed H ∈(−¹₂,¹₂). We note that Eqs. (3) are just the Euler–Lagrange equations of the

“H-surface functional”E^H:H^1,2(B,R³)∩L^∞(B,R³)→R, given by E^H(X):=

B

|∇X|²+4H

3 X, Xu∧Xvdudv. (5)

For some fixed polygonΓ andH∈(−¹₂,¹₂)Heinz [7] constructed a map ψ:T →

C1^∗(Γ ), · H^1,2∩C⁰

on a convex, open, bounded subsetT ⊂R^N (whereN +3 was the number of vertices ofΓ), whose following crucial properties shall be proved in this paper using Heinz’ isoperimetric inequality in [8], his boundary regularity theorem in [10] combined with an idea of Hildebrandt [12] and the author’s generalizations ([15] resp. [16]) of Courant’s fundamental ideas in [2,3]:

Reduction theorem.

(i) ψandf :=E^H◦ψare continuous onT. (ii) We have evenf=E^H◦ψ∈C¹(T ,R).

(iii) For every sequence{τⁿ}n∈Nwith dist(τⁿ, ∂T )→0 there holds f (τⁿ)→ ∞ forn→ ∞.

(iv) The restriction

ψ|K(f ):K(f )→^∼⁼ HH(Γ ) (6)

yields a homeomorphism between the compact setK(f )of critical points off and(HH(Γ ), · H^1,2∩C⁰).

We furthermore define the two following notions in

Definition 1.1. (i) A compact subsetP⊆HH(Γ )is termed a part (ofHH(Γ )) ifPhas an open neighborhoodUin (C₁^∗(Γ ), · H^1,2∩C⁰)which satisfiesP=U∩HH(Γ ), i.e. which separatesPfrom the complementHH(Γ )\P, and we set

KH(Γ ):=

P⊆HH(Γ )|P is a part

. (7)

(ii) AnE^H-minimizing partP (= ∅)ofHH(Γ )is characterized by the two following additional properties:

(1) E^H(X)≡const.(P) ∀X∈P,

(2) there exists an open neighborhoodU ofP in(C₁^∗(Γ ), · H^1,2∩C⁰)such that E^H(X)const.(P) ∀X∈U.

Now we can state the

Main theorem. To any closed polygonΓ ⊂D³andH∈(−¹₂,¹₂)one can assign an (H-surface-) index I: K_H(Γ )→Z

with the following properties:

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(i) I(HH(Γ ))=1, independent ofHandΓ.

(ii) Iis additive on K_H(Γ ), i.e. any decompositionP=_m

j=1Pj∈K_H(Γ ) (m1)yields I(P)=

m

j=1

I(Pj). (8)

(iii) Moreover the ˇCech cohomologyH (ˇ P)of anE^H-minimizing partP⊆HH(Γ )is non-trivial at most in degrees 0, . . . , N−1 and there even finitely generated,Pconsists of only finitely many connected components and its H-surface-index coincides with its ˇCech–Euler-characteristic:

I(P)=

N−1 i=0

(−1)ⁱdim_QHˇⁱ(P,Q)=: ˇχ (P,Q). (9)

This theorem immediately implies the following Corollaries.

(i) HH(Γ )= ∅.

(ii) A variant of the mountain-pass-lemma: If there existm2 differentE^H-minimizing partsPj inHH(Γ )with _j^m₌₁χ (ˇ Pj,Q)=1, then the complementHH(Γ )\_m

j=1Pj is not empty. This situation is encountered espe- cially if there existm2 homotopy equivalentE^H-minimizing partsPjinHH(Γ ). If in particular these parts Pj are points, i.e. isolated H-surfacesX₁, . . . , X_mwhich are local minimizers ofE^H in(C₁^∗(Γ ), · H^1,2∩C⁰), then the H-surface-index of the complementary partP:=HH(Γ )\ {X₁, . . . , X_m}amounts to

I(P)=1−m <0.

The author would like to point out the similarity of statement (i) of the main theorem to the result of Tromba’s papers [21] resp. [22], where Tromba constructs a “minimal surface index” which can be assigned to isolated minimal surfaces spanning a wire Γ ∈H^r,2(∂B,Rⁿ) (r >18, n3) that are non-degenerate in two different senses depending on the two casesn >3 resp.n=3. The fundamental tool for his “Index formula” of [21,22]

yields the deep Index Theorem of Böhme and himself [1] which guarantees that at least for an open, dense subset of boundary curvesΓ inH^r,2(∂B,Rⁿ)(so-called generic curves) the setM(Γ )of all minimal surfaces spanningΓ indeed consists of isolated and non-degenerate points in the two casesn >3 resp.n=3. One should also compare our main theorem to Struwe’s achievements in [19], where he develops a complete Morse theory for the description ofM(Γ ), providedΓ ⊂Rⁿ(n2) is aC⁵-regular boundary curve that spans only minimal surfaces which are non-degenerate critical points of the Dirichlet integral, which is guaranteed at least for generic wiresΓ ⊂Rⁿfor n4 by the Index Theorem [1] of Böhme and Tromba. Finally one should notice the similarity of Corollary (ii) to Struwe’s resp. Imbusch’s mountain-pass-lemma in [20], p. 51, resp. [14], p. 17, which also does not require the existence of different strict local minimizers of the considered functional (there it is the Dirichlet integral) on the class of admitted surfaces spanned into the boundary curve.

2. Fundamental properties ofE^H and H-surfaces

ForX∈H^1,2(B,R³)∩L^∞(B,R³)and anyL²-measurable subsetB⊆Bwe set DB(X):=1

2

B

|∇X|²dudv, FB(X):=

B

X, X_u∧X_vdudv and

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E_B^H(X):=2DB(X)+4H

3 FB(X) for|H|<1

2, (10)

and we shall use the abbreviationsD(X):=DB(X),F(X):=FB(X)andE^H(X):=E_B^H(X). At first we have for X∈H^1,2(B,R³)∩L^∞(B,R³)withXL^∞(B)1 due to|Xu∧Xv|¹₂|∇X|²the estimates

1−2 3|H|

2DB(X)E_B^H(X) 1+2 3|H|

2DB(X) (11)

on anyL²-measurable subsetB⊆B. By these inequalities one obtains as in Lemma 3.4 in [11] the lower semi- continuity ofE^H with respect to weak convergence inH^1,2(B,R³):

Lemma 2.1. For a sequence{Xj} ⊂H^1,2(B,R³)∩L^∞(B,R³)withXj XinH^1,2(B,R³)andXjL^∞(B)1 we have

E^H(X)lim inf

j→∞ E^H(X_j).

Analogously one can prove the same statement for the functionalG^H :=D+^4H₃ F(for|H|<1/2). By these tools one easily infers (see [17], Lemma 2.2)

Lemma 2.2. Let{Xj}be a sequence inH^1,2(B,R³)∩L^∞(B,R³)withXjL∞(B)1∀j∈N,

E^H(X_j)→E^H(X) and (12)

Xj X inH^1,2(B,R³) forj→ ∞, (13)

for some surfaceX∈H^1,2(B,R³), then there holds:

D(Xj)→D(X) forj→ ∞.

Now we consider the Dirichlet problem for E^H and prescribed boundary valuesr:=X|∂B of a surface X∈ C₁^∗(Γ ), i.e. the variational problem of minimizingE^H within a given boundary value class

[X|∂B] :=

Y∈H^1,2(B,R³)∩L^∞(B,R³)|Y−X∈H˚^1,2(B,R³), YL^∞(B)1 :

℘_H(X|∂B):E^H →Min. in[X|∂B]. (14)

By [11], Theorem 3.6 resp. 3.7, we have the following existence, uniqueness and regularity result:

Theorem 2.1. For anyX∈C₁^∗(Γ )andH∈(−¹₂,¹₂)there exists a unique solutionX^∗:=X^∗(X, H )∈ [X|∂B]of the problem℘_H(X|∂B)which additionally belongs toC⁰(B,R³)∩C²(B,R³)and solves Eq. (3) in the classical sense.

Leth(r)denote the uniquely determined harmonic extensionh∈C⁰(B,Rⁿ)∩C²(B,Rⁿ)of prescribed continuous boundary valuesr∈C⁰(∂B,Rⁿ)withrC⁰(∂B)1, forn1. From [9], Hilfssatz 5, we have the following boundary estimate for “small” solutions of (3):

Lemma 2.3. Let |H|< ¹₂ be arbitrarily fixed. For surfaces X ∈C²(B,R³)∩C⁰(B,R³) satisfying (3) and X_C⁰₍_B)1 there holds

X(ρe^iϑ)−X(e^iϑ) |H| 2(1−2|H|)h

|r|²

(ρe^iϑ)−r(e^iϑ)²+ 1− |H|

1−2|H|h(r)(ρe^iϑ)−r(e^iϑ) (15) for anyρ∈ [0,1]andϑ∈ [0,2π], wherer:=X|∂B.

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We use this boundary estimate to prove the following central compactness result:

Theorem 2.2. A sequence{X_k} ⊂C₁^∗(Γ )∩C²(B,R³)of surfaces which solve (3) and have bounded Dirichlet integrals, i.e.D(X_k)const.,∀k∈N, is compact inC⁰(B,R³).

Proof. On account of Hilfssatz 3 in [9] one can estimate the moduli of the gradients {|∇Xk|} on Bρ(0)⊂B uniformly by a constantC(ρ, H )for any fixedρ∈(0,1), which yields

Xk(w1)−Xk(w2)C(ρ, H )|w1−w2| ∀k∈N, (16)

∀w₁, w₂∈Bρ(0). Since theXk satisfyD(Xk)const.,Xk_C⁰(B)1 and the three-point-condition (2)∀k∈N we obtain by the Courant–Lebesgue lemma and Arzelà–Ascoli’s theorem a subsequence of the boundary values Xk|∂B=:rk (which will be renamed{rk}) that converges inC⁰(∂B,R³)to some continuous functionr. Conse- quently by the weak maximum principle for harmonic functions we infer for the unique harmonic extensionsh(r_k) resp.h(|r_k|²):

maxB

h

|r_k|²

−h

|r|²=max

∂B

|r_k|²− |r|²→0 and maxB

h(r_k)−h(r)√ 3 max

∂B |r_k−r| →0,

fork→ ∞, implying the equicontinuity of{h(r_k)}and{h(|r_k|²)}onB. Now combining this with Lemma 2.3 we obtain for a fixedH∈(−¹₂,¹₂): for an arbitrarily chosen >0 there is aδ(₃) >0 such that

X_k(ρe^iϑ)−X_k(e^iϑ) |H| 2(1−2|H|)h

|r_k|²

(ρe^iϑ)−r_k(e^iϑ)² + 1− |H|

1−2|H|h(rk)(ρe^iϑ)−rk(e^iϑ)<

3 (17)

if 1−ρ < δ(₃), uniformly∀k∈N. Together with the equicontinuity of theX_k on every closed discB_ρ(0)⊂B by (16) and the equicontinuity of the boundary valuesr_k on∂B one finally achieves the equicontinuity of{X_k} onB(see the proof of Theorem 2.2 in [17]). Hence byX_k_C⁰₍_B)1∀k∈NArzelà–Ascoli’s theorem yields the assertion. 2

Moreover we will use the following boundary regularity result for H-surfaces and an asymptotic expansion of the complex gradientX_u−iX_vabout a boundary branch point due to Heinz [10], Satz 3:

Theorem 2.3. LetX∈H^1,2(B,R³)∩C⁰(B,R³)∩C²(B,R³)be a solution of (3) and (4) which maps an open, connected arcγ⊂∂Bweakly monotonically into a regular, open Jordan curveΓof classC³.

(i) Then for everyw0∈γ there is an0>0 such that X∈C^1,ν(Z_w₀_,₀,R³) ∀ν∈(0,1)

on the closure of the “circular bigon”Z_w₀_,₀:=B₀(w₀)∩B.

(ii) Ifw₀∈γis a boundary branch point ofXand|X_u| ≡0, then one has (X_u−iX_v)(w)=a(w−w₀)^k+o

|w−w₀|^k

forZ_w₀_,₀w→w₀, for some complex vectora∈C³\ {0}and a positive integerk∈N.

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Using this theorem Hildebrandt derived in [12], Satz 3, the following

Corollary 2.1. The boundary values of an H-surfaceX∈HH(Γ )perform a homeomorphism:

X|∂B:∂B→^∼⁼ Γ.

For surfacesX∈H^1,2(B,R³)we denote its area by A(X):=

B

|Xu∧Xv|dudv.

In (39) we will make use of Heinz’ isoperimetric inequality for “small” H-surfaces, Theorem 3 in [8]:

Theorem 2.4. For an arbitrary “small” H-surfaceX∈HH(Γ )there holds A(X) 1

4π 1+h

1−hL(Γ )², (18)

whereh:= |H|XC⁰(B)(<¹₂)andL(Γ ):= length ofΓ.

3. Heinz’ mapψ

In this section we construct Heinz’ mapψ:T →(C^∗₁(Γ ), · H^1,2∩C⁰), where the setT ⊂R^N is defined as follows:

Definition 3.1. LetT be the set ofN-tuples (τ1, τ2, . . . , τN)=:τ∈(0,2π )^N⊂R^N which satisfy the following chain of inequalities:

0=ψ₀< τ₁<· · ·< τl< ψ₁< τl+1<· · ·< τm< ψ₂< τm+1<· · ·< τN<2π, (19) wherelandmare the same fixed indices as in (1).

ObviouslyT is a convex, open and bounded subset ofR^N.

Definition 3.2. To eachτ ∈T we assign a setU(τ )of surfacesX∈C₁^∗(Γ )which meet the following “Courant- condition”:

X|∂B(e^iτ^j)=A_j forj=1, . . . , N. (20)

At first for any fixedτ∈T one can easily construct boundary valuesryieldingh(r)∈U(τ ), hence we have Lemma 3.1.U(τ )= ∅ ∀τ∈T.

As we requireΓ to be a closed polygon one easily verifies the convexity ofU(τ ) for anyτ ∈T, which is a rather important point.

Now we state a slight generalization of Lemma 1 in [7] which Heinz asserted without proof (see Lemma 3.2 in [17] for a proof):

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Lemma 3.2. For any two surfacesX∈H^1,2(B,R³)and Z∈C⁰(B,R³)∩H^1,2(B,R³)and any monotonically increasing sequence of radiir_n^∗1 there exists in each interval[r_n^∗−_n, r_n^∗+_n], for_n:=(1−r_n^∗)/2, a subset RnwithL¹(Rn)_n, such that for an arbitrary sequence of radii{ξ_n}withξ_n∈Rn∀n∈Nthere holds:

2π 0

Z(ξneîϕ)−Z(eîϕ)Xϕ(ξneîϕ)dϕ→0 forn→ ∞.

The following basic integral identity, (1.9) in [7], is proved in [17], Lemma 3.3.

Lemma 3.3. For anyX, Z∈C⁰(B,R³)∩H^1,2(B,R³)there holds:

FBr(0)(X+Z)−FBr(0)(X)

=3

B_r(0)

Z, X_u∧X_vdudv+

B_r(0)

3X+Z, Z_u∧Z_vdudv+1 r

∂B_r(0)

X, Z∧(X_ϕ−Z_ϕ)

ds (21)

for a.e.r∈(0,1).

Combining Lemma 3.2 with the identity (21) Heinz achieved in [7], Lemma 2, the following formula (see also Lemma 3.4 in [17] for a more detailed proof):

Lemma 3.4. For any two surfacesX1, X2∈U(τ ), for an arbitraryτ∈T, we have:

F(X₂)−F(X₁)

=3

B

X₂−X₁, (X₁)_u∧(X₁)_v dudv+

B

2X₁+X₂, (X₂−X₁)_u∧(X₂−X₁)_v

dudv. (22)

Recalling the convexity of the setsU(τ )the above lemma yields the following crucial inequality due to Heinz [7], Lemma 3 (see also Lemma 3.5 in [17]):

Lemma 3.5. Letτ∈T be arbitrarily chosen, then for any two surfacesX₁, X₂∈U(τ )there holds:

1 2

E^H(X₁)+E^H(X₂)

−E^H X₁+X₂ 2

1

2

1−2|H|

D(X₁−X₂). (23)

Now we are prepared to prove the main result of this section using Theorems 2.1, 2.2 and the above inequality (see also Lemma 8 in [7]):

Proposition 3.1. For an arbitraryτ∈T there exists inU(τ )a uniquely determined minimizerX(τ )ofE^H, i.e.

E^H X(τ )

= inf

U(τ )E^H. (24)

FurthermoreX(τ )belongs toC²(B,R³)and solves (3) in the classical sense.

Proof. Existence: Let{Xk} ⊂U(τ )be a minimizing sequence forE^H, i.e.

E^H(Xk)→ inf

U(τ )E^H fork→ ∞. (25)

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Now Theorem 2.1 guarantees the existence of a sequence{X_k^∗} ⊂C⁰(B,R³)∩C²(B,R³)which satisfiesX_k^∗∈ [Xk|∂B],E^H(X^∗_k)=inf_[X_k|∂B]E^H and (3), thus{X^∗_k}is again a minimizing sequence forE^H inU(τ )in particular.

By (11) we see

D(X^∗_k)const. ∀k∈N, (26)

which together withX^∗_k_C⁰(B)1 impliesX^∗_kH^1,2(B)const.∀k∈N. Consequently we obtain a subsequence {X_k^∗

n}with X_k^∗

n X^∗ inH^1,2(B,R³) (27)

for someX^∗∈H^1,2(B,R³), which by the theorems of Rellich and Riesz implies the existence of a further subse- quence (that will be renamed{X^∗_k

n}) with X_k^∗

n(w)→X^∗(w) for a.e.w∈B.

On account of (26) Theorem 2.2 finally yields a further subsequence (that will be renamed{X^∗_k

n}again) satisfying X_k^∗

n→X^∗ inC⁰(B,R³). (28)

Hence, we obtainX^∗∈U(τ ). Together with lim_n_→∞E^H(X^∗_k

n)=inf_U_{(τ )}E^H and the weak lower semicontinuity of E^H by Lemma 2.1 applied to (27) we finally obtainE^H(X^∗)=inf_U_{(τ )}E^H. Furthermore together with[X^∗|∂B]= [X^∗|∂B] ∩C⁰(B,R³)⊂U(τ )and Remark 2.1 we infer:

E^H(X^∗)= inf

U(τ )E^H inf

[X^∗|∂B]E^H= inf

[X^∗|∂B]E^HE^H(X^∗),

thusX^∗is a solution of the Dirichlet problem℘_H(X^∗|∂B). Consequently Theorem 2.1 yields thatX^∗belongs to C²(B,R³)and solves (3) in the classical sense.

Uniqueness: LetX^∗₁ andX₂^∗be two E^H-minimizers inU(τ ), i.e. we have E^H(X^∗₁)=inf_U_{(τ )}E^H=E^H(X₂^∗).

AsU(τ )is convex ¹₂(X₁^∗+X₂^∗)belongs toU(τ ), thusE^H((X^∗₁+X₂^∗)/2)inf_U_{(τ )}E^H. Consequently Lemma 3.5 yields

1 2

1−2|H|

D(X^∗₁−X^∗₂) inf

U(τ )E^H−E^H X₁^∗+X₂^∗ 2

0,

henceD(X₁^∗−X^∗₂)=0 due to|H|<¹₂. SinceX^∗₁andX^∗₂are continuous onBand satisfy the same three-point- (and even Courant-) condition (2) we provedX^∗₁≡X^∗₂onB. 2

Now the above proposition suggests the following

Definition 3.3. For an arbitrarily chosen polygonΓ ⊂D³andH∈(−¹₂,¹₂)we define “Heinz’ map”

ψ:=ψ^Γ(H ):T →

C₁^∗(Γ ), · H^1,2∩C⁰

byψ (τ ):=X(τ ) (29)

andf:=f^Γ(H ):T →Rbyf :=E^H◦ψ.

4. Proof of the reduction theorem

On account of the invariance of the functional F with respect to orientation preserving diffeomorphisms φ:B∼= B, the “positive definiteness” ofE^H onC₁^∗(Γ )(11), Lemmas 2.1 and 2.2, Theorem 2.2 and Proposition 3.1 the assertion (i) of the reduction theorem can be proved exactly as Theorem 6.6 in [16] (see also [7], Lemma 10).

Since orientation preserving diffeomorphisms and especially conformal automorphisms of the disc will play a central role in later sections we add a few observations on such mappings.

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4.1. Inner variations of the discB

Definition 4.1. A family of orientation preservingC³-diffeomorphismsφ:=φ(·, ):B→ B,∈ [−₀, ₀](for a fixed, sufficiently small₀>0), are called inner variations ofBof “medium” type if they are three times continuously differentiable inon[−₀, ₀], i.e.φ∈C³(B× [−₀, ₀],R²)and a perturbation of the identity:φ₀=id_B. The set of these variations ofBwill be denoted byV.

Remark 4.1. (a) A familyφ∈V possesses a Taylor expansion with respect to:φ=id_B+λ+o()onB, for →0, whereλ:=_∂^∂ φ|=0is called the generator of the familyφ, and its remainder of first order can be estimated by¹₂φC²².

(b) The inverse family{φ⁻¹}is again a family of inner variations ofBof medium type and possesses the Taylor expansionφ⁻¹ =id_B−λ+o()onB, for →0.

Remark 4.2. Now we consider the action of inner variationsφ∈V on the set T. LetKT be compact and φ∈V arbitrarily chosen. Then there is an₀=₀(K, φ) >0 such that the family{φ}∈[−₀,₀] induces a flow (φ):K→T via arg(·):S¹∼=R/(2πZ)by:

τ=(τ₁, . . . , τ_N)→

arg◦φ|∂B(e^iτ¹), . . . ,arg◦φ|∂B(e^iτ^N)

=:(φ)(τ ). (30)

Next we consider the mapΛ:V×T →R^Ngiven byΛ(φ, τ ):=_∂^∂ (φ)(τ )|=0. A simple calculation yields Λ(φ, τ )= ∂

∂(φ)(τ ) =0

= 1

ie^iτ¹λ(e^iτ¹), . . . , 1

ie^iτ^Nλ(e^iτ^N)

,

whereλdenotes the generator of the family{φ}. Now we fix someτ∈T. For an arbitraryΥ =(Υ₁, . . . , Υ_N)∈ S^N⁻¹we want to construct aφ∈Vsuch that the pair(φ, τ )is mapped byΛontoΥ. This can easily be carried out by means of “hill functions”h_m∈C_c^∞(R²) (m=1, . . . , N )with respect to the fixedτ =(τ₁, . . . , τ_N)which can explicitly be given by

h_m(w):=

e¹^re⁻^1/(r^−|^w⁻êîτm^|⁾ for|w−eîτ^m|< r,

0 for|w−e^iτ^m|r

form∈ {1, . . . , N}, where we have putτN+1:=τ₁and require r <1

2

min

l∈{1,...,N},k∈{0,1,2}

|eîτ^l−eîτ^l⁺¹|,|eîτ^l−eîψ^k| .

Obviously thehmhave the following properties:

h_m(e^iτ^l)=δ_ml form, l∈ {1, . . . , N}, (31)

h_m(e^iψ^k)=0 form=1, . . . , Nandk=0,1,2.

Now we setυ:=_N

m=1h_mΥ_monB. Then the inner variations φ(w):=wexp(iυ(w)), which we term(τ, Υ )- variations, induce

(φ)(τ )=(τ₁+Υ₁, . . . , τ_N+Υ_N)=τ+Υ ∈T (for||sufficiently small), which completes the construction.

Remark 4.3. Finally we mention that in general a family{φ} ∈V affects the three points{e^iψ^k}k=0,1,2; consequently the inner variationsX◦φ of some surfaceX∈C₁^∗(Γ )might leave this class. In order to overcome this

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technical difficulty we follow Courant’s idea of [2] to “renorm” {φ}by a family of conformal automorphisms {K} ⊂Aut(B)which is uniquely determined by its desired property

φ˜(eîψ^k):=(φ◦K)(eîψ^k)=^! eîψ^k fork=0,1,2.

Moreover via a straightforward calculation one has to ensure that this renormed family{ ˜φ}is an element ofV again (see [15], p. 71, for a proof).

4.2. Proof of the points (ii)–(iv) of the reduction theorem

The proof of statement (ii) of the reduction theorem is based on the following result (see [17], Lemma 4.4, for a proof).

Proposition 4.1. Let{φ}∈[−₀,₀]be a family inV andX∈H^1,2(B,Rⁿ)some surface, then D(X◦φ):[−₀, ₀] →R

is twice continuously differentiable with respect toand we have (i):

d

dD(X◦φ) =0

=1 2

B

a(λ¹_u−λ²_v)+b(λ¹_v+λ²_u)dudv=:∂D(X, λ), (32) wherea:= |X_u|²− |X_v|²andb:=2X_u, X_vand (ii):

E^H(X◦φ)E^H(X)+2∂D(X, λ)+²cD(X), (33)

where the constant c depends only on{φ}(and{φ⁻¹}) but not onX.

On account of the inequalities (33) and (11) the following theorem can be proved as Theorem 6.13 in [16], using the continuity off, Lemma 6.12 in [16] and Remark 4.1(b) (see also Lemma 11 in [7]).

Theorem 4.1. (i) At every point τ ∈T there exist all directional derivatives of f. (ii) Letτ ∈T andΥ ∈S^N⁻¹ be arbitrarily chosen and consider the corresponding(τ, Υ )-variationsφ(w):=exp(iυ(w))(see Remark 4.2), then there holds

∂

∂Υf (τ )=2∂D

X(τ ),−λ

, (34)

whereλ(w):=iwυ(w)forw∈ B.

Now the assertion (ii) of the reduction theorem can be proved as Theorem 6.14 in [16] combining the above theorem with Lemma 6.12 in [16] (in [7] this statement is asserted in Theorem 1 without proof).

By (11) assertion (iii) of the reduction theorem can be proved as Lemma 6.16 in [16].

Now using f ∈C¹(T ) the proof of Theorem 4.1 yields the following counterpart of the statement of Theo- rem 4.1(ii) (see Corollary 6.15 in [16] for a proof).

Corollary 4.1. Consider someφ∈V which leaves the three points{e^iψ^k}invariant, its generatorλ, some fixed pointτ∈T and the corresponding vectorΥ :=Λ(φ, τ )=_∂^∂ (φ)(τ )|=0(see Remark 4.2). Then

∂D

X(τ ), λ

= −1 2

∂

∂Υf (τ )= −1 2

∇f (τ ), Υ

. (35)

A combination of this representation of inner variations of the Dirichlet integral with Remark 4.3 and Lemma 6.18, Proposition 6.19 in [16] yields analogously to Theorem 6.17 in [16] (see also Theorem 2 in [7]):

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Corollary 4.2. Ifτ∈T is a critical point off, thenX(τ )is conformally parametrized onB, i.e.X(τ )satisfies (4).

Now recalling Proposition 3.1 the above corollary just states that the restriction ofψto the setK(f )of critical points off maps intoHH(Γ ), i.e.

im(ψ|K(f ))⊂HH(Γ ). (36)

In order to achieve even

im(ψ|K(f ))=HH(Γ ) (37)

one needs Lemma 6 in [7] (see also Theorem 4.3 in [17] for a more detailed proof):

Theorem 4.2. If an H-surfaceXbelongs to the intersectionHH(Γ )∩U(τ )for someτ ∈T, then there holds E^H(X)= inf

U(τ )E^H. (38)

Now let there be given an arbitrary H-surfaceX∈HH(Γ ). Then it must be contained in some class U(τ ) for some τ ∈T due to the surjectivity of its boundary values onto Γ. Hence, combining the above theorem with Proposition 3.1 we expose X to be ψ (τ ), which means im(ψ )⊃HH(Γ ). Furthermore by (32) we see

∂D(X, λ)=0∀λ∈C¹(B,R²)sinceXis conformally parametrized onB, i.e. sinceXsatisfies (4). Consequently by Theorem 4.1(ii)τ must be a critical point off, which implies even im(ψ|K(f ))⊃HH(Γ ). Thus together with (36) we finally obtain (37). Furthermore Corollary 2.1 implies

Corollary 4.3.

ψ|K(f ):K(f )HH(Γ ) is injective.

Proof. If we assume the contrary, i.e. that there exist points{τ¹=τ²} ⊂K(f )withX(τ¹)=ψ (τ¹)=ψ (τ²)= X(τ²), then especially the boundary valuesX(τ¹)|∂B=X(τ²)|∂B∈C⁰(∂B,R³)would have to coincide. However in the tuples(τ₁¹, . . . , τ_N¹)=τ¹=τ²=(τ₁², . . . , τ_N²)there is at least one different pairτ_j¹=τ_j²(e.g.τ_j¹< τ_j²), and due to the weak monotonicity of the boundary valuesX(τ¹)|∂B=X(τ²)|∂B these would have to be constant on the open, connected arc between exp(iτ_j¹)and exp(iτ_j²), in contradiction to Corollary 2.1. 2

Moreover combiningA(X)=D(X)for any X∈HH(Γ )with (37), (11) and Heinz’ isoperimetric inequality (18) we obtain for every critical pointτ∈K(f )the estimate

0f (τ )=E^H ψ (τ )

<const.

|H|,L(Γ )

=:c. (39)

Together with point (iii) of the reduction theorem this yields K(f )f⁻¹

[0, c)

T . (40)

Hence,K(f )reveals to be a compact subset ofT, which together with the continuity ofψ, (37) and Corollary 4.3 implies assertion (iv) of the reduction theorem.

5. Application of Dold’s fixed point theory and ˇCech cohomology

In this section we combine the reduction theorem with singular homology, ˇCech cohomology [6] and Dold’s fixed point theory [4] in order to define the H-surface-indexI: K_H(Γ )→Zand derive its asserted properties.

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5.1. Definition of the H-surface indexI

We define a continuous flowφ:T × [−t₀, t₀] →T by

φ(τ, t):=τ−t η(τ )∇f (τ ) (41)

for a sufficiently smallt₀>0, depending on∇f and a cut-off functionη∈C_c⁰(T ,[0,1])to be defined precisely in Definition 5.1 below with the property

f⁻¹ [0, c]

[η=1], (42)

wherec=c(|H|,L(Γ ))is the constant from (39). We compute d

dtf φ(τ, t)

= −η(τ )

∇f φ(τ, t)

,∇f (τ )

(43)

∀t∈ [−t₀, t₀]and∀τ ∈T. At first we state the following elementary

Lemma 5.1. If there holds |∇f|δ for some δ >0 on a compact subset K⊂T, then there exists some t= t(K,∇f )∈(0, t₀]such that

f φ(τ, t)

f (τ )− t

2η(τ )δ²f (τ ) (44)

uniformly∀τ∈Kand∀t∈ [0, t]and f

φ(τ, t)

f (τ )+|t|

2 η(τ )δ²f (τ ) (45)

uniformly∀τ∈Kand∀t∈ [−t,0].

Proof. From|∇f|δ >0 onK, the compactness ofK, the continuity of∇f and (43) one can easily derive the existence of somet∈(0, t₀](depending onKand∇f) such that

d dtf

φ(τ, t)

−η(τ )

2 ∇f (τ )²

uniformly∀τ ∈Kand∀t∈ [−t, t]. Hence, by the fundamental theorem of calculus we obtain:

f φ(τ, t)

−f (τ )= t

0

d dtf

φ(τ, t)

dt−tη(τ )

2 ∇f (τ )²−tη(τ ) 2 δ²

∀τ∈Kand∀t∈ [0, t]and

f φ(τ, t)

−f (τ )= t

0

d dtf

φ(τ, t) dt= −

0 t

d dtf

φ(τ, t)

dt|t|η(τ )

2 ∇f (τ )²|t|η(τ ) 2 δ²

∀τ∈Kand∀t∈ [−t,0]. 2

From the above inequalities we derive

Corollary 5.1. There is somet^∗∈(0, t₀]such that φ(·, t )|_f−1([0,c]):f⁻¹

[0, c]

→f⁻¹ [0, c]

and

φ(·, t )|f⁻¹([0,c])id_f−1([0,c]) relK(f ) (46)