93(2013) 471-530

MINIMIZERS OF THE WILLMORE FUNCTIONAL UNDER FIXED CONFORMAL CLASS

Ernst Kuwert & Reiner Sch¨atzle

Abstract

We prove the existence of a smooth minimizer of the Willmore
energy in the class of conformal immersions of a given closed Rie-
mann surface into IR^{n}, n= 3,4, if there is one conformal immer-
sion with Willmore energy smaller than a certain bound W^{n,p}
depending on codimension and genus p of the Riemann surface.

For tori in codimension 1, we know W^{3}^{,}^{1}= 8π.

1. Introduction

For an immersed closed surface f : Σ→IR^{n} the Willmore functional
is defined by

W(f) = 1 4

Z

Σ

|H~|^{2} dµ_{g},

where H~ denotes the mean curvature vector of f, g=f^{∗}geuc the pull-
back metric and µ_{g} the induced area measure on Σ . For orientable
Σ of genus p, the Gauß equations and the Gauß-Bonnet theorem give
rise to equivalent expressions

(1.1) W(f) = 1 4

Z

Σ

|A|^{2} dµg+ 2π(1−p) = 1
2

Z

Σ

|A^{0}|^{2} dµg+ 4π(1−p)
where A denotes the second fundamental form, A^{0} =A−^{1}_{2}g⊗H its
tracefree part.

We always have W(f) ≥4π with equality only for round spheres, see [Wil82] in codimension one that is n= 3 . On the other hand, if W(f) <8π then f is an embedding by an inequality of Li and Yau in [LY82].

E. Kuwert and R. Sch¨atzle were supported by the DFG Forschergruppe 469, and R. Sch¨atzle was supported by the Hausdorff Research Institute for Mathematics in Bonn. Main parts of this article were discussed during a stay of both authors at the Centro Ennio De Giorgi in Pisa.

Received 06/01/2011.

471

Critical points of W are called Willmore surfaces or more precisely Willmore immersions. They satisfy the Euler-Lagrange equation which is the fourth order, quasilinear geometric equation

∆_{g}H~ +Q(A^{0})H~ = 0

where the Laplacian of the normal bundle along f is used and Q(A^{0})
acts linearly on normal vectors along f by

Q(A^{0})φ:=g^{ik}g^{jl}A^{0}_{ij}hA^{0}_{kl}, φi.

The Willmore functional is scale invariant and moreover invariant under
the full M¨obius group of IR^{n} . As the M¨obius group is non-compact,
this invariance causes difficulties in the construction of minimizers by
the direct method.

In [KuSch11], we investigated the relation of the pull-back metric
g to constant curvature metrics on Σ after dividing out the M¨obius
group. More precisely, we proved in [KuSch11] Theorem 4.1 that for
immersions f : Σ→IR^{n}, n= 3,4, Σ closed, orientable and of genus
p=p(Σ)≥1 satisfying W(f)≤ Wn,p−δ for someδ >0 , where

(1.2)

W3,p= min 8π,4π+P

k(β_{p}^{3}_{k}−4π) | P

kp_{k}=p,0≤p_{k}< p
,
W4,p = min 8π, β_{p}^{4}+ 8π/3,4π+P

k(β_{p}^{4}_{k}−4π) |
P

kp_{k}=p,0≤p_{k}< p
,
and

(1.3)

β_{p}^{n}:= inf{W(f) |f : Σ→IR^{n} immersion, p(Σ) =p,Σ orientable},
there exists a M¨obius transformation Φ of the ambient space IR^{n}
such that the pull-back metric ˜g= (Φ◦f)^{∗}g_{euc} differs from a constant
curvature metric e^{−2u}g˜ by a bounded conformal factor, more precisely

kukL^{∞}(Σ),k ∇ukL^{2}(Σ,˜g)≤C(p, δ).

In this paper, we consider conformal immersions f : Σ→IR^{n} of a fixed
closed Riemann surface Σ and prove existence of smooth minimizers
in this conformal class under the above energy assumptions.

Theorem 7.3Let Σ be a closed Riemann surface of genus p≥1
with smooth conformal metric g_{0} with

W(Σ, g_{0}, n) :=

inf{W(f) |f : Σ→IR^{n} smooth immersion conformal to g_{0} }<Wn,p,
where Wn,p is defined in (1.2) and n= 3,4 .

Then there exists a smooth conformal immersion f : Σ → IR^{n}
which minimizes the Willmore energy in the set of all smooth conformal
immersions. Moreover f satisfies the Euler-Lagrange equation

(1.4) ∆gH~ +Q(A^{0})H~ =g^{ik}g^{jl}A^{0}_{ij}q_{kl} on Σ,

where q is a smooth transverse traceless symmetric 2-covariant tensor
with respect to the Riemann surface Σ , that is with respect to g =
f^{∗}g_{euc} .

For the proof, we select by the compactness theorem [KuSch11]

Theorem 4.1 a minimizing sequence of smooth immersions f_{m} : Σ →
IR^{n} conformal to g0 , that is

W(f_{m})→ W(Σ, g_{0}, n),
converging

f_{m} →f weakly inW^{2,2}(Σ, IR^{n}),

where the limitf defines a uniformly positive definite pull-back metric
g:=f^{∗}g_{euc} =e^{2u}g_{0}

with u ∈L^{∞}(Σ) . We call such f aW^{2,2}−immersion uniformly con-
formal to g_{0} .

For such f we can define the Willmore functional and obtain by lower semicontinuity

W(f)≤ lim

m→∞W(fm) =W(Σ, g0, n).

Now there are two questions on f :

• Is f a minimizer in the larger class of uniformly conformal
W^{2,2}−immersions?

• Is f smooth?

For the first question we can approximate uniformly conformal f by
[SU83] strongly in W^{2,2} by smooth immersions ˜f_{m} , but these ˜f_{m}
are in general not conformal to g_{0} anymore. To this end, we develop a
correction in conformal class and get smooth conformal immersions ¯fm

converging strongly in W^{2,2} to f . We obtain that the infimum over
smooth conformal immersions and over uniformly conformal immersions
coincide.

Theorem 5.1 For a closed Riemann surface Σ of genus p ≥ 1
with smooth conformal metric g_{0}

W(Σ, g0, n) =

= inf{W(f) |f : Σ→IR^{n} smooth immersion conformal to g0 }

= inf{W(f) |f : Σ→IR^{n} is a W^{2,2}−immersion uniformly
conformal to g_{0} }.

Smoothness of f is obtained by the usual whole filling procedure as in the proof of the existence of minimizers for fixed genus in [Sim93].

Here again we need the correction in conformal class in an essential way.

Our method applies to prove smoothness of any uniformly conformal
W^{2,2}−minimizer.

Theorem 7.4Let Σ be a closed Riemann surface of genus p≥1
with smooth conformal metric g_{0} and f : Σ → IR^{n} be a uniformly
conformal W^{2,2}−immersion, that is g = f^{∗}g_{euc} = e^{2u}g_{0} with u ∈
L^{∞}(Σ), which minimizes the Willmore energy in the set of all smooth
conformal immersions

W(f) =W(Σ, g_{0}, n),

then f is smooth and satisfies the Euler-Lagrange equation

∆_{g}H~ +Q(A^{0})H~ =g^{ik}g^{jl}A^{0}_{ij}q_{kl} on Σ,

where q is a smooth transverse traceless symmetric 2-covariant tensor with respect to g .

Finally to explain the correction in conformal class, we recall by
Poincar´e’s theorem, see [Tr], that any smooth metric g on Σ is confor-
mal to a unique smooth unit volume constant curvature metric g_{poin}=
e^{−2u}g . Denoting by Met the set of all smooth metrics on Σ , by
Met_{poin} the set of all smooth unit volume constant curvature metrics
on Σ , and by D:={Σ−→^{≈} Σ} the set of all smooth diffeomorphisms
of Σ , we see that Met_{poin}/D is the moduli space of all conformal
structures on Σ . Actually to deal with a space of better analytical
properties, we consider D0 :={φ∈ D | φ≃id_{Σ} } and the Teichm¨uller
space T :=Metpoin/D0. This is a smooth open manifold of dimension
2 for p = 1 and of dimension 6p−6 for p ≥ 2 and the projection
π :Met → T is smooth, see [FiTr84] and [Tr]. We define in §3 the
first variation in Teichm¨uller space of an immersion f : Σ→ IR^{n} and
a variation V ∈C^{∞}(Σ, IR^{n}) by

0 = d

dtπ((f+tV)^{∗}g_{euc})|t=0=δπ_{f}.V
and consider the subspace

Vf :=δπ_{f}.C^{∞}(Σ, IR^{n})⊆T_{π(f}^{∗}_{g}_{euc}_{)}T.

In case of Vf = T_{π}T , which we call f of full rank in Teichm¨uller
space, the correction in Teichm¨uller space is easily achieved by implicit
function theorem.

A severe problem arises in the degenerate case, when f is not of full rank in Teichm¨uller space. In this case, we prove in Proposition 4.1 that

dim Vf = dim T −1,

g^{ik}g^{jl}A^{0}_{ij}q_{kl}= 0

for some non-zero smooth transverse traceless symmetric 2-covariant tensor q with respect to g, and f is isothermic locally around all but finitely many points of Σ , that is around all but finitely many points of Σ there exist local conformal principal curvature coordinates. Then by implicit function theorem, we can do the correction in Vf . In the one dimensional orthogonal complement spanned by some e⊥ Vf , we do the correction with the second variation by monotonicity. To deal with the positive and the negative part along span{e} , we need two variations V± satisfying

±hδ^{2}π_{f}(V_{±}), ei>0, δπˆ_{f}.V_{±}= 0,

whoose existence is established in a long computation in Proposition 4.2.

Recently, Kuwert and Li and independently Riviere have extended
the existence of non-smooth conformally constrained minimizers in any
codimension, where branch points may occur if W(Σ, g_{0}, n) ≥8π , see
[KuLi10] and [Ri10]. In [Ri10] the obtained conformally constrained
minimizers are smooth and satisfy the Euler-Lagrange equation (1.4)
in the full rank case, whereas in the degenerate case these are proved
to be isothermic, but the smoothness and the verification of (1.4) as in
Theorem 7.4 are left open. Everything that is done in this article for
W(Σ, g_{0}, n) < Wn,p forn = 3,4, can be done with [Sch12] Theorem
4.1 for W(Σ, g_{0}, n)<8π for any n.

Acknowledgement. The main ideas of this paper came out during a stay of both authors at the Centro Ennio De Giorgi in Pisa. Both authors thank very much the Centro Ennio De Giorgi in Pisa for the hospitality and for providing a fruitful scientific atmosphere.

2. Direct method

Let Σ be a closed, orientable surface of genus p≥1 with smooth metric g satisfying

W(Σ, g, n) :=

inf{W(f)| f : Σ→IR^{n} smooth immersion conformal tog }<Wn,p,
as in the situation of Theorem 7.3 where Wn,p is defined in (1.2) above
and n= 3,4 . To get a minimizer, we consider a minimizing sequence
of immersions f_{m}: Σ→IR^{n} conformal to g in the sense

W(f_{m})→ W(Σ, g, n).

After applying suitable M¨obius transformations according to [KuSch11]

Theorem 4.1 we will be able to estimate f_{m} inW^{2,2}(Σ) and W^{1,∞}(Σ) ,
see below, and after passing to a subsequence, we get a limit f ∈
W^{2,2}(Σ)∩W^{1,∞}(Σ) which is an immersion in a weak sense, see (2.6)

below. To prove that f is smooth, which implies that it is a minimizer,
and that f satisfies the Euler-Lagrange equation in Theorem 7.3 we
will consider variations, say of the form f+V . In general, these are not
conformal to g anymore, and we want to correct it by f+V +λ_{r}V_{r}
for suitable selected variations V_{r} . Now even these are not confor-
mal to g since the set of conformal metrics is quite small in the set
of all metrics. To increase the set of admissible pull-back metrics, we
observe that it suffices for (f +V +λ_{r}V_{r})◦φ being conformal to g
for some diffeomorphism φof Σ . In other words, the pullback metric
(f+V +λ_{r}V_{r})^{∗}g_{euc} need not be conformal to g , but has to coincide
only in the modul space. Actually, we will consider the Teichm¨uller
space, which is coarser than the modul space, but is instead a smooth
open manifold, and the bundle projection π :Met → T of the sets of
metrics Met into the Teichm¨uller space T is smooth, see [FiTr84],
[Tr]. Clearly W(Σ, g, n) depends only on the conformal structure de-
fined by g, in particular it descends to Teichm¨uller space and leads to
the following definition.

Definition 2.1. We define Mn,p :T →[0,∞] forp≥1, n≥3, by selecting a closed, orientable surface Σ of genus p and

Mn,p(τ) := inf{W(f)|f : Σ→IR^{n} smooth immersion, π(f^{∗}g_{euc}) =τ }.
We see

Mn,p(π(g)) =W(Σ, g, n).

Next, inf_{τ}Mn,p(τ) =β_{p}^{n} for the infimum under fixed genus defined in
(1.3), and, as the minimum is attained and 4π < β_{p}^{n}<8π , see [Sim93]

and [BaKu03],

4π <min

τ∈T Mn,p(τ) =β_{p}^{n}<8π.

In the following proposition, we consider a slightly more general situa- tion than above.

Proposition 2.2. Let f_{m} : Σ → IR^{n}, n = 3,4, be smooth immer-
sions of a closed, orientable surface Σof genus p≥1 satisfying

(2.1) W(f_{m})≤ Wn,p−δ

and

(2.2) π(f_{m}^{∗}g_{euc})→τ_{0} in T.

Then replacing f_{m} by Φ_{m}◦f_{m}◦φ_{m} for suitable M¨obius transformations
Φ_{m} and diffeomorphisms φ_{m} of Σ homotopic to the identity, we get

(2.3) lim sup

m→∞ kfmkW^{2,2}(Σ)<∞,

and f_{m}^{∗}g_{euc} = e^{2u}^{m}g_{poin,m} for some unit volume constant curvature
metrics g_{poin,m} with

(2.4) ku_{m}kL^{∞}(Σ),k ∇u_{m} kL^{2}(Σ,gpoin,m)<∞,
g_{poin,m} →g_{poin} smoothly
with π(g_{poin}) =τ_{0} . After passing to a subsequence

(2.5) f_{m} →f weakly in W^{2,2}(Σ),weakly^{∗} in W^{1,∞}(Σ),
u_{m} →u weakly inW^{1,2}(Σ),weakly^{∗} in L^{∞}(Σ),
and

(2.6) f^{∗}g_{euc}=e^{2u}g_{poin}

where (f^{∗}g_{euc})(X, Y) :=h∂_{X}f, ∂_{Y}fi for X, Y ∈TΣ.

Proof. Clearly, replacing fm by Φm◦fm◦φm as above does neither change the Willmore energy nor the projection into the Teichm¨uller space.

By [KuSch11] Theorem 4.1 after applying suitable M¨obius transfor-
mations, the pull-back metric g_{m} := f_{m}^{∗}g_{euc} is conformal to a unique
constant curvature metric e^{−2u}^{m}gm =:gpoin,m of unit volume with

oscΣum,k ∇um kL^{2}(Σ,gm)≤C(p, δ).

Combining the M¨obius transformations with suitable homotheties, we
may further assume that f_{m} has unit volume. This yields

1 = Z

Σ

dµ_{g}_{m} =
Z

Σ

e^{2u}^{m} dµ_{g}_{poin,m},

and, as g_{poin,m} has unit volume as well, we conclude that u_{m} has a
zero on Σ , hence

kum kL^{∞}(Σ),k ∇um kL^{2}(Σ,gpoin,m)≤C(p, δ)
and, as f_{m}^{∗}geuc=gm =e^{2u}^{m}gpoin,m ,

k ∇f_{m}kL^{∞}(Σ,gpoin,m)≤C(p, δ).

Next

∆gpoin,mfm =e^{2u}^{m}∆gmfm=e^{2u}^{m}H~_{f}_{m}
and

k∆_{g}_{poin,m}f_{m} k^{2}_{L}^{2}_{(Σ,g}_{poin,m}_{)}≤e^{2 max}^{u}^{m}
Z

Σ

|H~_{f}_{m}|^{2}e^{2u}^{m} dµ_{g}_{poin,m}

(2.7) = 4e^{2 max}^{u}^{m}W(f_{m})≤C(p, δ).

To get further estimates, we employ the convergence in Teichm¨uller space (2.2). We consider a slice S of unit volume constant curvature

metrics for τ_{0} ∈ T , see [FiTr84], [Tr]. There exist unique ˜g_{poin,m} ∈
S withπ(˜g_{poin,m}) =π(g_{m})→τ_{0} for m large enough, hence

φ^{∗}_{m}g˜poin,m =gpoin,m

for suitable diffeomorphisms φm of Σ homotopic to the identity. Re-
placing f_{m} byf_{m}◦φ_{m} , we get g_{poin,m} = ˜g_{poin,m} ∈ S and

g_{poin,m}→g_{poin} smoothly

with g_{poin}∈ S, π(g_{poin}) =τ_{0}. Then translating f_{m} suitably, we obtain
kf_{m}kL^{∞}(Σ)≤C(p, δ).

Moreover standard elliptic theory, see [GT] Theorem 8.8, implies by (2.7),

kfmkW^{2,2}(Σ)≤C(p, δ, gpoin)
for m large enough.

Selecting a subsequence, we get f_{m} → f weakly inW^{2,2}(Σ), f ∈
W^{1,∞}(Σ), Df_{m} →Df pointwise almost everywhere, u_{m} →uweakly in
W^{1,2}(Σ), pointwise almost everywhere, and u ∈ L^{∞}(Σ) . Putting
g:=f^{∗}g_{euc} , that is g(X, Y) :=h∂_{X}f, ∂_{Y}fi, we see

g(X, Y)← h∂_{X}f_{m}, ∂_{Y}f_{m}i=g_{m}(X, Y)

=e^{2u}^{m}gpoin,m(X, Y)→e^{2u}gpoin(X, Y)

for X, Y ∈TΣ and pointwise almost everywhere on Σ , hence
f^{∗}g_{euc}=g=e^{2u}g_{poin}.

q.e.d.

We call a mapping f ∈W^{1,∞}(Σ, IR^{n}) withg:=f^{∗}g_{euc}=e^{2u}g_{poin}, u∈
L^{∞}(Σ), g_{poin} a smooth unit volume constant curvature metric on Σ as
in (2.6) a lipschitz immersion uniformly conformal to g_{poin} or in short a
uniformly conformal lipschitz immersion. If further f ∈W^{2,2}(Σ, IR^{n}) ,
we call f a W^{2,2}−immersion uniformly conformal to gpoin or a
uniformly conformal W^{2,2}−immersion. In this case, we see g, u ∈
W^{1,2}(Σ) , hence we can define weak Christoffel symbols Γ ∈ L^{2} in
local charts, a weak second fundamental form A and a weak Riemann
tensor R via the equations of Weingarten and Gauß

∂ijf = Γ^{k}_{ij}∂_{k}f+Aij,
R_{ijkl}=hA_{ik}, A_{jl}i − hA_{ij}, A_{kl}i.

Of course this defines H, A~ ^{0}, K for f as well. Moreover we define the
tangential and normal projections for V ∈W^{1,2}(Σ, IR^{n})∩L^{∞}(Σ, IR^{n})

cπ_{f}.V :=g^{ij}h∂if, Vi∂jf,

π_{f}^{⊥}.V :=V −π_{Σ}.V ∈W^{1,2}(Σ, IR^{n})∩L^{∞}(Σ, IR^{n}).

(2.8)

Mollifing f as in [SU83]§4 Proposition, we get smooth f_{m}: Σ→IR^{n}
with

(2.9) f_{m}→f strongly in W^{2,2}(Σ),weakly^{∗} in W^{1,∞}(Σ)
and, as Df ∈W^{1,2} , that locally uniformly

sup

|x−y|≤C/m

d(Df_{m}(x), Df(y))→0.

This implies for that the pull-backs are uniformly bounded from below and above

(2.10) c_{0}g≤f_{m}^{∗}g_{euc}≤Cg

for some 0< c_{0} ≤C <∞ and m large, in particular f_{m} are smooth
immersions.

3. The full rank case

For a smooth immersion f : Σ→IR^{n} of a closed, orientable surface
Σ of genus p ≥ 1 and V ∈ C^{∞}(Σ, IR^{n}) , we see that the variations
f+tV are still immersions for small t. We put g_{t}:= (f+tV)^{∗}g_{euc}, g=
g_{0} =f^{∗}g_{euc} and define

(3.1) δπ_{f}.V :=dπg.∂tgt|t=0.
The elements of

Vf :=δπ_{f}.C^{∞}(Σ, IR^{n})⊆T_{πg}T

can be considered as the infinitesimal variations of g in Teichm¨uller space obtained by ambient variations of f . We call f of full rank in Teichm¨uller space, if dimVf = dim T . In this case, the necessary corrections in Teichm¨uller space mentioned in§2 can easily be achieved by the inverse function theorem, as we will see in this section.

Writing g=e^{2u}g_{poin} for some unit volume constant curvature metric
g_{poin} by Poincar´e’s theorem, see [Tr] Theorem 1.3.7, we see π(g_{t}) =
π(e^{−2u}g_{t}) , hence for an orthonormal basis q^{r}(g_{poin}), r= 1, . . . ,dim T,
of transvere traceless tensors in S_{2}^{T T}(gpoin) with respect to gpoin

(3.2) δπ_{f}.V =dπ_{g}.∂_{t}g_{t|t=0} =dπ_{g}_{poin}.e^{−2u}∂_{t}g_{t|t=0}

=

dim T

X

r=1

he^{−2u}∂_{t}g_{t|t=0}.q^{r}(g_{poin})igpoindπ_{g}_{poin}.q^{r}(g_{poin}).

Calculating in local charts

(3.3) g_{t,ij} =g_{ij}+th∂_{i}f, ∂_{j}Vi+th∂_{j}f, ∂_{i}Vi+t^{2}h∂_{i}V, ∂_{j}Vi,

we get

he^{−2u}∂tgt|t=0, q^{r}(gpoin)igpoin

= Z

Σ

g_{poin}^{ik} g_{poin}^{jl} e^{−2u}∂_{t}g_{t,ij}_{|t=0}q^{r}_{kl}(g_{poin}) dµ_{g}_{poin}

= 2 Z

Σ

g^{ik}g^{jl}h∂_{i}f, ∂_{j}Viq_{kl}^{r}(g_{poin}) dµ_{g}

= −2 Z

Σ

g^{ik}g^{jl}h∇^{g}_{j}∇^{g}_{i}f, Viq_{kl}^{r}(gpoin) dµg

−2 Z

Σ

g^{ik}g^{jl}h∂_{i}f, Vi∇^{g}_{j}q_{kl}^{r}(g_{poin}) dµ_{g}

= −2 Z

Σ

g^{ik}g^{jl}hA^{0}_{ij}, Viq_{kl}^{r}(g_{poin}) dµ_{g},

as q ∈S_{2}^{T T}(gpoin) = S_{2}^{T T}(g) is divergence- and tracefree with respect
to g. Therefore

δπ_{f}.V =

dim T

X

r=1

2 Z

Σ

g^{ik}g^{jl}h∂_{i}f, ∂_{j}Viq^{r}_{kl}(g_{poin}) dµ_{g} dπ_{g}_{poin}.q^{r}(g_{poin})

(3.4) =

dim T

X

r=1

−2 Z

Σ

g^{ik}g^{jl}hA^{0}_{ij}, Viq_{kl}^{r}(g_{poin}) dµ_{g} dπ_{g}_{poin}.q^{r}(g_{poin}),
and δπ_{f} and Vf are well defined for uniformly conformal lipschitz im-
mersions f and V ∈W^{1,2}(Σ) .

First, we select variations whoose image via δπ_{f} form a basis of Vf .
Proposition 3.1. For a uniformly conformal lipschitz immersion f
and finitely many points x_{0}, . . . , x_{N} ∈Σ , there exist V_{1}, . . . , V_{dim}_{V}_{f} ∈
C_{0}^{∞}(Σ− {x0, . . . , xN}, IR^{n}) such that

(3.5) Vf =span {δπ_{f}.V_{s} |s= 1, . . . ,dim Vf }.

Proof. Clearly δπ_{f} :C^{∞}(Σ, IR^{n})→ T_{πg}_{poin}T is linear. For suitable
W ⊆C^{∞}(Σ, IR^{n}) , we obtain a direct sum decomposition

C^{∞}(Σ, IR^{n}) =ker δπ_{f} ⊕W

and see that δπ_{f}|W →im δπ_{f} is bijective, hence dim W = dim Vf

and a basis V_{1}, . . . , V_{dim}_{V}_{f} of W satisfies (3.5).

Next we choose cutoff functions ϕ_{̺}∈C_{0}^{∞}(∪^{N}_{k=0}B_{̺}(x_{k})) such that 0≤
ϕ_{̺}≤1, ϕ_{̺}= 1 on ∪^{N}k=0B_{̺/2}(x_{k}) and |∇ϕ_{̺}|g ≤C̺^{−1} , hence 1−ϕ_{̺}∈

C_{0}^{∞}(Σ−{x_{0}, . . . , x_{N}}),1−ϕ_{̺}→1 on Σ−{x_{0}, . . . , x_{N}},R

Σ|∇ϕ_{̺}|g dµ_{g} ≤
C̺ → 0 for̺ →0 . Clearly ϕ_{m}V_{s} ∈C_{0}^{∞}(Σ− {x_{0}, . . . , x_{N}}, IR^{n}) and
by (3.4)

δπ_{f}.(ϕ_{m}V_{s})→δπ_{f}.V_{s},

hence ϕ_{m}V_{1}, . . . , ϕ_{m}V_{dim}_{V}_{f} satisfies for large m all conclusions of the

proposition. q.e.d.

We continue with a convergence criterion for the first variation.

Proposition 3.2. Let f : Σ→ IR^{n} be a uniformly conformal lips-
chitz immersion approximated by smooth immersions f_{m} with pull-back
metrics g = f^{∗}g_{euc} = e^{2u}g_{poin}, g_{m} = f_{m}^{∗}g_{euc} = e^{2u}^{m}g_{poinm} for some
smooth unit volume constant curvature metrics g_{poin}, g_{poin,m} and sat-
isfying

(3.6)

fm→f weakly in W^{2,2}(Σ),weakly^{∗} in W^{1,∞}(Σ),
Λ^{−1}g_{poin}≤g_{m}≤Λg_{poin},

ku_{m} kL^{∞}(Σ)≤Λ
for some Λ<∞ . Then for any W ∈L^{2}(Σ, IR^{n})

π_{f}_{m}.W →π_{f}.W
as m→ ∞ .

Proof. By (3.6)

(3.7)

g_{m}=f_{m}^{∗}g_{euc}→f^{∗}g_{euc}=g weakly inW^{1,2}(Σ),
weakly^{∗} inL^{∞}(Σ),

Γ^{k}_{g}_{m}_{,ij} →Γ^{k}_{g,ij} weakly inL^{2},
A_{f}_{m}_{,ij}, A^{0}_{f}_{m}_{,ij} →A_{f,ij}, A^{0}_{f,ij} weakly inL^{2},

R

Σ

|K_{g}_{m}| dµ_{g}_{m}≤ ^{1}_{2}R

Σ

|A_{f}_{m}|^{2} dµ_{g}_{m} ≤C,
hence by [FiTr84], [Tr],

(3.8) q^{r}(g_{poin,m})→q^{r}(g_{poin})

weakly inW^{1,2}(Σ),
weakly^{∗} inL^{∞}(Σ),
dπ_{g}_{poin,m}.q^{r}(g_{poin,m})→dπ_{g}_{poin}.q^{r}(g_{poin}).

Then by (3.4)

δπ_{f}_{m}.W

=

dim T

X

r=1

−2 Z

Σ

g_{m}^{ik}g^{jl}_{m}hA^{0}_{f}_{m}_{,ij}, Wiq_{kl}^{r}(gpoin,m) dµgm dπgpoin,m.q^{r}(gpoin,m)

→δπ_{f}.W.

q.e.d.

In the full rank case, the necessary corrections in Teichm¨uller space mentioned in §2 are achieved in the following lemma by the inverse function theorem.

Lemma 3.3. Let f : Σ → IR^{n} be a uniformly conformal lipschitz
immersion approximated by smooth immersions f_{m} satisfying (2.2) -
(2.6).

If f is of full rank in Teichm¨uller space, then for arbitrary x_{0} ∈
Σ, neighbourhood U_{∗}(x_{0}) ⊆Σ of x_{0} andΛ < ∞ , there exists a neigh-
bourhood U(x_{0})⊆U_{∗}(x_{0}) of x_{0} , variations V_{1}, . . . , V_{dim}_{T} ∈C_{0}^{∞}(Σ−
U(x_{0}), IR^{n}) , satisfying (3.5), and δ >0, C < ∞, m_{0} ∈ IN such that
for any V ∈C_{0}^{∞}(U(x_{0}), IR^{n}) with f_{m}+V a smooth immersion for
some m≥m_{0} , and V = 0 or

(3.9)

Λ^{−1}g_{poin}≤(f_{m}+V)^{∗}g_{euc}≤Λg_{poin},
kV kW^{2}^{,}^{2}(Σ)≤Λ,

R

U∗(x0)

|A_{f}_{m}_{+V}|^{2} dµ_{f}_{m}_{+V} ≤ε_{0}(n),
where ε0(n) is as in Lemma A.1, and any τ ∈ T with

(3.10) d_{T}(τ, τ_{0})≤δ,

there exists λ∈IR^{dim}^{T} satisfying

π((f_{m}+V +λ_{r}V_{r})^{∗}g_{euc}) =τ
and

|λ| ≤Cd_{T}

π((f_{m}+V)^{∗}g_{euc}), τ
.

Proof. By (2.3), (2.4) and Λ large enough, we may assume
(3.11) ku_{m}, Df_{m} kW^{1}^{,}^{2}(Σ)∩L^{∞}(Σ)≤Λ,

Λ^{−1}g_{poin}≤f_{m}^{∗}g_{euc}≤Λg_{poin},
in particular

(3.12)

Z

Σ

|A_{f}_{m}|^{2} dµ_{f}_{m} ≤C(Σ, gpoin,Λ).

Putting ν_{m} := |∇^{2}gpoinf_{m}|^{2}gpoinµ_{g}_{poin} , we see ν_{m}(Σ)≤C(Λ, g_{poin}) and
for a subsequence νm → ν weakly^{∗} inC_{0}^{0}(Σ)^{∗} . Clearly ν(Σ)< ∞ ,
and there are at most finitely many y_{1}, . . . , y_{N} ∈Σ withν({y_{i}})≥ε_{1} ,
where we choose ε_{1} >0 below.

As f is of full rank, we can select V_{1}, . . . , V_{dim}_{T} ∈ C_{0}^{∞}(Σ −
{x_{0}, y_{1}, . . . , y_{N}}, IR^{n}) withspan{δπ_{f}.V_{r}}=T_{g}_{poin}T by Proposition 3.1.

We choose a neighbourhood U_{0}(x_{0}) ⊆ U_{∗}(x_{0}) ofx_{0} with a chart
ϕ0 :U0(x0)−→^{≈} B2(0), ϕ0(x0) = 0 ,

supp V_{r}∩U_{0}(x_{0}) =∅ forr = 1, . . . ,dimT,

put x_{0} ∈U_{̺}(x_{0}) =ϕ^{−1}_{0} (B_{̺}(0)) for 0< ̺≤2 and choose x_{0} ∈U(x_{0})⊆
U_{1/2}(x_{0}) small enough, as we will see below.

Next for any x ∈ ∪^{dim}r=1^{T}supp V_{r} , there exists a neighbourhood
U0(x) of x with a chart ϕx : U0(x) −→^{≈} B2(0), ϕx(x) = 0, U0(x) ∩
U0(x0) =∅, ν(U0(x))< ε1 and in the coordinates of the chart ϕx

(3.13)

Z

U0(x)

g^{ik}_{poin}g_{poin}^{jl} g_{poin,rs}Γ^{r}_{g}_{poin}_{,ij}Γ^{s}_{g}_{poin}_{,kl} dµ_{g}_{poin} ≤ε_{1}.

Putting x∈U_{̺}(x) =ϕ^{−1}_{x} (B_{̺}(0))⊂⊂U_{0}(x) for 0< ̺≤2 , we see that
there are finitely many x_{1}, . . . , x_{M} ∈ ∪^{dim}r=1^{T}supp V_{r} such that

∪^{dim}r=1^{T}supp V_{r} ⊆ ∪^{M}k=1U_{1/2}(x_{k}).

Then there exists m0∈IN such that for m≥m0

Z

U1(x_{k})

|∇^{2}gpoinf_{m}|^{2}gpoin dµ_{g}_{poin} < ε_{1} fork= 1, . . . , M.

For V andm≥m_{0} as above, we put ˜f_{m,λ} :=f_{m}+V +λ_{r}V_{r} . Clearly
supp(f_{m}−f˜_{m,λ})⊆ ∪^{M}k=0U_{1/2}(x_{k}).

By (3.9), (3.11), (3.13), and |λ|< λ_{0} ≤1/4 small enough independent
of m and V , ˜f_{m,λ} is a smooth immersion with

(3.14) (2Λ)^{−1}g_{poin}≤g˜_{m,λ}:= ˜f_{m,λ}^{∗} g_{euc}≤2Λg_{poin},
and if V 6= 0 by (3.9) and the choice of U_{0}(x) that

Z

U1(xk)

|A_{f}_{˜}

m,λ|^{2} dµ_{g}_{˜}_{m,λ}≤ε_{0}(n) fork= 0, . . . , M

for C(Λ, g_{poin})(ε_{1}+λ_{0})≤ε_{0}(n) . If V = 0 , we see supp(f_{m}−f˜_{m,λ})⊆

∪^{M}_{k=1}U_{1/2}(x_{k}) . Further by (3.12)
Z

Σ

|K_{g}_{˜}_{m,λ}|dµ_{˜}_{g}_{m,λ} ≤ 1
2

Z

Σ

|A_{f}_{˜}

m,λ|^{2} dµ_{˜}_{g}_{m,λ}

≤ 1 2

Z

Σ

|A_{f}_{m}|^{2} dµ_{f}_{m}+1
2

M

X

k=0

Z

U_{1/2}(x_{k})

|A_{f}_{˜}

m,λ|^{2} dµ_{g}_{˜}_{m,λ}

≤C(Σ, g_{poin},Λ) + (M+ 1)ε_{0}(n).

This verifies (A.3) and (A.4) for f = f_{m},f˜ = ˜f_{m,λ}, g_{0} = g_{poin} and
different, but appropriate Λ . (A.2) follows from (2.4) and (3.12). Then
for the unit volume constant curvature metric ˜g_{poin,m,λ} =e^{−2˜}^{u}^{m,λ}˜g_{m,λ}
conformal to ˜g_{m,λ} by Poincar´e’s theorem, see [Tr] Theorem 1.3.7, we
get from Lemma A.1 that

(3.15) ku˜_{m,λ} kL^{∞}(Σ),k ∇u˜_{m,λ} kL^{2}(Σ,gpoin)≤C
with C <∞ independent of m andV .

From (3.9), we have a W^{2,2}∩W^{1,∞}−bound on ˜f_{0} , hence for ˜f_{m,λ}.
On Σ−U(x_{0}) , we get f˜_{m,λ} = f_{m}+λ_{r}V_{r} → f weakly inW^{2,2}(Σ−
U(x_{0})) and weakly^{∗} inW^{1,∞}(Σ −U(x_{0})) form_{0} → ∞, λ_{0} → 0 by
(2.5). If V = 0 , then f˜_{m,λ} = f_{m} +λ_{r}V_{r} → f weakly in W^{2,2}(Σ)
and weakly^{∗} inW^{1,∞}(Σ) form_{0} → ∞, λ_{0} →0 by (2.5). Hence letting
m0→ ∞, λ0 →0, U(x0)→ {x0}, we conclude

(3.16) f˜_{m,λ}→f weakly inW^{2,2}(Σ),weakly^{∗} inW^{1,∞}(Σ),
in particular

˜

g_{m,λ} →f^{∗}g_{euc}=e^{2u}g_{poin}=:g

weakly inW^{1,2}(Σ),
weakly^{∗} inL^{∞}(Σ).

Together with (3.14), this implies by [FiTr84], [Tr],
(3.17) π(˜g_{m,λ})→τ_{0}.

We select a chart ψ : U(π(g_{poin})) ⊆ T → IR^{dim}^{T} and put ˆπ :=

ψ◦π, δˆπ =dψ◦dπ,Vˆf :=dψ_{πg}_{poin}.Vf . By (3.17) for m_{0} large enough,
λ_{0} and U(x_{0}) small enough independent of V , we get π(˜g_{m,λ})∈U(τ_{0})
and define

Φ_{m}(λ) := ˆπ(˜g_{m,λ}).

This yields by (3.4), (3.14), (3.15), (3.16) and Proposition 3.2
DΦm(λ) = (δˆπf˜_{m,λ}.Vr)r=1,...,dimT →(δˆπ_{f}.Vr)r=1,...,dimT

=:A∈IR^{dim}^{T ×dim}^{T}.

As span{δπ_{f}.V_{r}} = T_{g}_{poin}T ∼= IR^{dim}^{T} , the matrix A is invertible.

Choosing m0 large enough, λ0 and U(x0) small enough, we obtain
kDΦm(λ)−Ak≤1/(2kA^{−1} k),

hence by standard inverse function theorem for any ξ ∈IR^{dim}^{T} with

|ξ−Φ_{m}(0)|< λ_{0}/(2kA^{−1}k) there exists λ∈B_{λ}_{0}(0) with
Φ_{m}(λ) =ξ,

|λ| ≤2kA^{−1} k |ξ−Φm(0)|.

As
d_{T}

π((f_{m}+V)^{∗}g_{euc}), τ

≤d_{T}(π(˜g_{0}), τ_{0}) +d_{T}(τ_{0}, τ)< d_{T}(π(˜g_{0}), τ_{0}) +δ
by (3.10), we see for δ small enough, m_{0} large enough and U(x_{0})
small enough by (3.17) that there exists λ∈IR^{dim}^{T} satisfying

π((f_{m}+V +λ_{r}V_{r})^{∗}g_{euc}) =π(˜g_{m,λ}) =τ,

|λ| ≤Cd_{T}

π((f_{m}+V)^{∗}g_{euc}), τ
,

and the lemma is proved. q.e.d.

4. The degenerate case

In this section, we consider the degenerate case when the immersion is not of full rank in Teichm¨uller space. First we see that the image in Teichm¨uller space looses at most one dimension.

Proposition 4.1. For a uniformly conformal immersion f ∈
W^{2,2}(Σ, IR^{n}) we always have

(4.1) dim Vf ≥dim T −1.

f is not of full rank in Teichm¨uller space if and only if
(4.2) g^{ik}g^{jl}A^{0}_{ij}q_{kl}= 0

for some non-zero smooth transverse traceless symmetric 2-covariant
tensor q with respect to g=f^{∗}g_{euc} .

In this case, f is isothermic locally around all but finitely many points of Σ , that is around all but finitely many points of Σ there exist local conformal principal curvature coordinates.

Proof. Let g_{poin} = e^{−2u}g be the unit volume constant curvature
metric conformal to g = f^{∗}geuc . For q ∈ S_{2}^{T T}(gpoin) , we put Λq :
C^{∞}(Σ, IR^{n})→IR

Λ_{q}.V :=−2
Z

Σ

g^{ik}g^{jl}hA^{0}_{ij}, Viq_{kl} dµ_{g}
and define the annihilator

Ann:={q ∈S_{2}^{T T}(gpoin) |Λq= 0 }.

As dπ_{g}_{poin}|S_{2}^{T T}(g_{poin}) → T_{g}_{poin}T is bijective, we see by (3.4) and
elementary linear algebra

(4.3) dim T = dim Vf + dim Ann.

Clearly,

(4.4) q∈Ann⇐⇒g^{ik}g^{jl}A^{0}_{ij}q_{kl}= 0,

which already yields (4.2) Choosing a conformal chart with respect to
the smooth metric g_{poin}, we see g_{ij} =e^{2v}δ_{ij} for somev∈W^{1,2}∩L^{∞}
and A^{0}_{11}=−A^{0}_{22}, A^{0}_{12}=A^{0}_{21}, q_{11}=−q_{22}, q_{12}=q_{21}, as both A^{0} andq ∈
S_{2}^{T T}(g_{poin}) are symmetric and tracefree with respect to g=e^{2u}g_{poin} .
This rewrites (4.4) into

(4.5) q∈Ann⇐⇒A^{0}_{11}q_{11}+A^{0}_{12}q_{12}= 0.

The correspondence between S_{2}^{T T}(g_{poin}) and the holomorphic qua-
dratic differentials is exactly that in conformal coordinates

(4.6) h:=q_{11}−iq_{12} is holomorphic.

Now if (4.1) were not true, there would be two linearly independent
q^{1}, q^{2} ∈ Ann by (4.3). Likewise the holomorphic functions h^{k} :=

q^{k}_{11}−iq_{12}^{k} are linearly independent over IR , in particular neither of
them vanishes identically, hence these vanish at most at finitely many
points, as Σ is closed. Then h^{1}/h^{2} is meromorphic and moreover not a
real constant. This implies that Im(h^{1}/h^{2}) does not vanish identically,
hence vanishes at most at finitely many points. Outside these finitely
many points, we calculate

Im(h^{1}/h^{2}) =|h^{2}|^{−2}Im(h^{1}h^{2}) =|h^{2}|^{−2}det

q^{1}_{11} q_{12}^{1}
q^{2}_{11} q_{12}^{2}

, hence

det

q^{1}_{11} q^{1}_{12}
q^{2}_{11} q^{2}_{12}

vanishes at most at finitely many points and by (4.5)

(4.7) A^{0} = 0.

Approximating f by smooth immersions as in (2.10), we get

∇kH~ = 2g^{ij}∇iA^{0}_{jk}= 0 weakly,

where ∇=D^{⊥} denotes the normal connection in the normal bundle
along f . Therefore |H~| is constant and

∂_{k}H~ =g^{ij}h∂_{k}H, ∂~ _{j}fi∂_{i}f =−g^{ij}hH, A~ _{jk}i∂_{i}f weakly.

Using ∆_{g}_{poin}f = e^{2u}∆_{g}f = e^{2u}H~ and u ∈ W^{1,2}∩L^{∞} , we conclude
successively that f ∈C^{∞} . Then (4.7) implies that f parametrizes a
round sphere or a plane, contradicting p≥1 , and (4.1) is proved.

Next if f is not of full rank in Teichm¨uller space, there exits
q ∈ Ann− {0} 6= ∅ by (4.3), and the holomorphic function h in
(4.6) vanishes at most at finitely many points. In a neighbourhood
of a point where h does not vanish, there is a holomorphic function
w with (w^{′})^{2} =ih . Then w has a local inverse z and using w as

new local conformal coordinates, h transforms into (h◦z)(z^{′})^{2} =−i,
hence q_{11}= 0, q_{12}= 1 in w−coordinates. By (4.5)

A_{12}= 0,

and w are local conformal principal curvature coordinates. q.e.d.

Since we loose at most one dimension in the degenerate case, we
will do the necessary corrections in Teichm¨uller space mentioned in§2
by investigating the second variation in Teichm¨uller space. To be more
precise, for a smooth immersion f : Σ→IR^{n} with pull-back metric g=
f^{∗}g_{euc} conformal to a unit volume constant curvature metric g_{poin} =
e^{−2u}f^{∗}g_{euc} by Poincar´e’s theorem, see [Tr] Theorem 1.3.7, we select
a chart ψ : U(π(g_{poin})) ⊆ T → IR^{dim}^{T} , put πˆ := ψ◦π,Vˆf :=

dψ_{πg}_{poin}.Vf , and define the second variation in Teichm¨uller space of f
with respect to the chart ψ by

(4.8) δ^{2}πˆ_{f}(V) :=d
dt

2

ˆ

π((f +tV)^{∗}geuc)_{|t=0}.

If f is not of full rank in Teichm¨uller space, we know by the previous
Proposition 4.1 that dim Vf = dim T −1 , and we do the corrections
in the remaining direction e⊥Vˆf,|e|= 1 , which is unique up to the
sign, by variations V_{±} satisfying

±hδ^{2}πˆ_{f}(V_{±}), ei>0.

We construct V_{±} locally around an appropriate x_{0} ∈ Σ in a chart
ϕ:U(x_{0})−→^{≈} B_{1}(0) of conformal principal curvature coordinates that
is

(4.9) g=e^{2v}g_{euc}, A_{12}= 0,

which exists by Proposition 4.1, and moreover assuming that x_{0} is not
umbilical that is

(4.10) A^{0}(x_{0})6= 0,

as Σ is not a sphere. We choose some unit normal N0 at f inf(0) ,
some η∈C_{0}^{∞}(B1(0)) and put η̺(y) :=η(̺^{−1}y) ,

V :=η_{̺}N0.

For ̺ small, the support of V is close to 0 and V is nearly normal.

Putting f_{t} :=f +tV, g_{t} :=f_{t}^{∗}g_{euc}, g =g_{0} , we see ˆπ(g_{t}) = ˆπ(e^{−2u}g_{t})
and calculate

δ^{2}πˆ_{f}(V) =d
dt

2

ˆ

π(e^{−2u}gt)_{|t=0}

=dˆπgpoin.(e^{−2u}(∂ttgt)_{|t=0}) +d^{2}πˆgpoin(e^{−2u}(∂tgt)_{|t=0}, e^{−2u}(∂tgt)_{|t=0}).

We will see that the second term, which is quadratic in g , is much smaller than the first one for ̺ small. To calculate the effect of the first

term on e, we consider the unique q_{e}∈S_{2}^{T T}(g) with dˆπ_{g}_{poin}.q_{e}(g_{poin}) =
e and see by (3.3) that

hdˆπ_{g}_{poin}.(e^{−2u}(∂_{tt}g_{t})_{|t=0}), ei

= Z

Σ

g^{ik}_{poin}g^{jl}_{poin}e^{−2u}(∂_{tt}g_{t,ij})_{|t=0}q_{e,kl} dµ_{g}_{poin}

= 2 Z

Σ

g^{ik}g^{jl}h∂_{i}V, ∂_{j}Viq_{e,kl} dµ_{g}.

Now q_{e} satisfies (4.2), hence by (4.9) and (4.10) that q_{e,11}=−q_{e,22}≡
0, q_{e,12}=q_{e,21}≡q∈IR− {0} inB_{1}(0) and

hδ^{2}πˆ_{f}(V), ei →4qe^{−2v(0)}
Z

B1(0)

∂_{1}η∂_{2}η dL^{2} for̺→0.

Now it just requires to find appropriate η ∈ C_{0}^{∞}(B_{1}(0)) to achieve
a positive and a negative sign. Our actual choice is given after (4.26)
below.

In the following long computation, we make the above considerations
work. Firstly we show that the second variation in Teichm¨uller space
is well defined for uniformly conformal W^{2,2}−immersions f and V ∈
W^{1,2}(Σ) . Then we estimate rigorously the lower order approximations
which were neglected above.

We proceed for smooth f and V by decomposing
(4.11) e^{−2u}(∂_{t}g_{t})_{|t=0}=σg_{poin}+LXg_{poin}+q,

with σ ∈C^{∞}(Σ), X ∈ X(Σ), q ∈S^{T T}_{2} (g_{poin}) and continue by recalling
{σg_{poin}+LXg_{poin}}=ker dˆπ_{g}_{poin} with

δ^{2}πˆ_{f}(V) = dˆπ_{g}_{poin}.(e^{−2u}(∂_{tt}g_{t})_{|t=0})

+d^{2}πˆ_{g}_{poin}(σg_{poin}+LXg_{poin}+q, σg_{poin}+LXg_{poin}+q)

= dˆπ_{g}_{poin}.(e^{−2u}(∂_{tt}g_{t})_{|t=0}) +d^{2}πˆ_{g}_{poin}(q, q)

+d^{2}πˆ_{g}_{poin}(σg_{poin}+LXg_{poin}, σg_{poin}+LXg_{poin}+ 2q)

= dˆπ_{g}_{poin}.

e^{−2u}(∂_{tt}g_{t})_{|t=0}− LXLXg_{poin}

−2σLXg_{poin}−2σq−2LXq

+d^{2}ˆπ_{g}_{poin}(q, q).