Willmore surfaces in spheres: the DPW approach via the conformal Gauss map

Abstract

The paper builds a DPW approach of Willmore surfaces via conformal Gauss maps. As applications, we provide descriptions of minimal surfaces in ${\mathbb {R}}^{n+2}$, isotropic surfaces in $S^4$ and homogeneous Willmore tori via the loop group method. A new example of a Willmore two-sphere in $S^6$ without dual surfaces is also presented.

Appendix: Two decomposition theorems

In this section we discuss the basic decomposition theorems for loop groups. We will use the notation introduced in Sect. 3. Since the decomposition theorems usually are proven for loops in simply-connected groups we will assume in this section that G is simply-connected and will therefore always use, to avoid confusion, the notation ${\tilde{G}}$. We would like to point out that the case of the group $G = SL(2,{\mathbb {R}})$ is included in our presentation, but needs, at places, some interpretation, since in this case ${\tilde{G}}$ is not a matrix group, while ${\tilde{G}}^{\mathbb {C}}= SL(2,{\mathbb {C}})$ is a matrix group and “contains” ${\tilde{G}}$ as a (non-isomorphic) image of the natural homomorphism.

1.1 Birkhoff decomposition

Starting from ${\tilde{G}}$ and an inner involution $\sigma$, there is a unique extension, denoted again by $\sigma$, to ${\tilde{G}}^{\mathbb {C}}$. The corresponding fixed point subgroups will be denoted by ${\tilde{K}}^{\mathbb {C}}$. Note that the latter group is connected, by a result of Springer−Steinberg.

Next we will consider the twisted loop group $\Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$. General loop group theory implies that, since we consider inner involutions only, we have $\Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma } \cong \Lambda {\tilde{G}}^{{\mathbb {C}}}$. On the other hand, we know $\pi _0 (\Lambda H) = \pi _1 (H)$ for any connected Lie group H, whence we infer that $\Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$ is connected. And since ${\tilde{K}}^{\mathbb {C}}$ is connected, also the groups $\Lambda ^{+} {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$ and $\Lambda ^{-} {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$ are connected.

For the loop group method used in this paper two decomposition theorems are of crucial importance. The first is

Theorem 5.1

(Birkhoff decomposition theorem) Let ${\tilde{G}}^{{\mathbb {C}}}$ denote a simply-connected complex Lie group with connected real form ${\tilde{G}}$ and let $\sigma$ be an inner involution of ${\tilde{G}}$ and ${\tilde{G}}^{{\mathbb {C}}}$. Then the following statements hold

• (1) $\Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma } = \bigcup \Lambda ^{-} {\tilde{G}}^{{\mathbb {C}}}_{\sigma } \cdot {\tilde{\omega }} \cdot \Lambda ^{+} {\tilde{G}}^{{\mathbb {C}}}_{\sigma },$ where the ${\tilde{\omega }}$’s are representatives of the double cosets.

• (2) The multiplication

  $$\begin{aligned} \Lambda ^-_* {\tilde{G}}^{{\mathbb {C}}}_{\sigma } \times \Lambda ^+ {\tilde{G}}^{{\mathbb {C}}}_{\sigma }\rightarrow \Lambda ^-_* {\tilde{G}}^{{\mathbb {C}}}_{\sigma } \cdot \Lambda ^+ {\tilde{G}}^{{\mathbb {C}}}_{\sigma } \end{aligned}$$

  (5.1)

  is a complex analytic diffeomorphism and the (left) “big cell” $\Lambda ^-_* {\tilde{G}}^{{\mathbb {C}}}_{\sigma } \cdot \Lambda ^+ {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$ is open and dense in $\Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$.

• (3) More precisely, every g in $\Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$ can be written in the form

  $$\begin{aligned} g=g_-\cdot {\tilde{\omega }}\cdot g_+ \end{aligned}$$

  (5.2)

  with $g_{\pm }\in \Lambda ^{\pm } {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$, and ${\tilde{\omega }} : S^1 \rightarrow {\tilde{T}} \subset Fix^\sigma ({\tilde{G}}^{\mathbb {C}})$ a homomorphism , where ${\tilde{T}}$ is a maximal compact torus in ${\tilde{G}}^{\mathbb {C}}$ fixed pointwise by $\sigma$.

Proof

The decomposition above has been proven for algebraic loop groups in [25]. Our results follow by completeness in the Wiener Topology (see e.g. [14]). $\square$

Remark 5.2

• (1) Our actual goal is to obtain a Birkhoff decomposition theorem for $(\Lambda G_\sigma ^{\mathbb {C}})^0$. The restriction to the connected component is possible, since we will always consider maps from connected surfaces into the loop group which attain the value I at some point of the surface.

  This is very fortunate: since we have obtained above a Birkhoff decomposition for the simply connected complexified universal group, we will attempt to obtain the desired Birkhoff decomposition by projection. Since $\Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$ is connected, applying the natural extension of the natural projection ${\tilde{\pi }}^{\mathbb {C}}: {\tilde{G}}^{{\mathbb {C}}} \rightarrow {G}^{{\mathbb {C}}}$ to $\Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$ we obtain as image the connected component $(\Lambda G_\sigma ^{\mathbb {C}})^0$ of $\Lambda G_{\sigma }^{\mathbb {C}}$ . We thus obtain the desired Birkhoff decomposition by projecting the terms on the right side. But since the groups $\Lambda ^{+} {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$ and $\Lambda ^{-} {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$ are connected their images under the projection are $\Lambda ^{+}_{\mathcal {C}}{G}^{{\mathbb {C}}}_{\sigma }$ and $\Lambda ^{-}_{\mathcal {C}}{G}^{{\mathbb {C}}}_{\sigma }$ respectively. From this the Birkhoff Decomposition Theorem for $(\Lambda G_\sigma ^{\mathbb {C}})^0$ follows. The special case of primary interest in this paper will be discussed in detail in the following remark.

• (2) Let J denote a nondegenerate quadratic form in ${\mathbb {R}}^{n+4}$ and $SO(J,{\mathbb {C}})$ the corresponding real special orthogonal group. Let $SO(J,{\mathbb {C}})$ denote the complexified special orthogonal group. Then $SO(J,{\mathbb {C}})$ is connected and has fundamental group $\pi _{1}(SO(J,{\mathbb {C}}))\cong {\mathbb {Z}}/2{\mathbb {Z}}$. Moreover, if $\sigma$ is an involutive inner automorphism, we have $\Lambda SO(J,{\mathbb {C}})\cong \Lambda SO(J,{\mathbb {C}})_{\sigma }$. Therefore

  $$\begin{aligned} \pi _0 ( \Lambda SO(J,{\mathbb {C}})_{\sigma }) \cong \pi _0 ( \Lambda SO(J,{\mathbb {C}}))\cong \pi _{1}(SO(J,{\mathbb {C}}))\cong {\mathbb {Z}}/2{\mathbb {Z}}. \end{aligned}$$

  (5.3)

  The loop group $\Lambda SO^+(1,{\tilde{n}},{\mathbb {C}})_{\sigma }$ thus has two connected components. Finally, choosing $\sigma , K$ and $K^{\mathbb {C}}$ in the Willmore setting, the group $K^{\mathbb {C}}$ has two connected components. Therefore also $\Lambda ^+ {G}^{{\mathbb {C}}}_{\sigma }$ and $\Lambda ^{-} {G}^{{\mathbb {C}}}_{\sigma }$ have two connected components.

• (3) Much of the above is contained in [33], Section 8.5 (see also [35]). Note, however, that our real group $G=SO^+(1,n+3)$ is not compact.

Proof of Theorem 3.1

Consider the universal cover $\pi : Spin(1,n+3, {\mathbb {C}})\rightarrow SO(1,n+3,{\mathbb {C}})$. Then $\pi$ induces a homomorphism from $\Lambda Spin(1,n+3, {\mathbb {C}})_\sigma$ to $(\Lambda SO(1,n+3, {\mathbb {C}})_\sigma )^0$, the identity component of $\Lambda SO(1,n+3, {\mathbb {C}})_\sigma$. Projecting the decomposition of Theorem 5.1 with ${{\tilde{G}}}^{\mathbb {C}}= Spin(1,n+3,{\mathbb {C}})$ to $SO(1,n+3, {\mathbb {C}})$, we obtain the Birkhoff factorization Theorem 3.1. $\square$

1.2 Iwasawa decomposition

From here on we will write, for convenience, $\Lambda G^0_\sigma$ for $(\Lambda G_\sigma )^0$. For our geometric applications we also need a second loop group decomposition. Ideally we would like to be able to write any $g\in (\Lambda G^{\mathbb {C}})^0_\sigma$ in the form $g=hv_+$ with $h\in (\Lambda G)^0_\sigma$ and $v_+ \in (\Lambda ^+ G^{\mathbb {C}})^0_\sigma = \Lambda ^+_{\mathcal {C}} G^{\mathbb {C}}_\sigma$. Unfortunately, this is not always possible.

For the discussion of this situation we start again by considering the universal cover ${\tilde{\pi }}^{\mathbb {C}}:{\tilde{G}}^{\mathbb {C}}\rightarrow G^{\mathbb {C}}$. Then $\tau$, the anti-holomorphic involution of $G^{\mathbb {C}}$ defining G, and $\sigma$ have natural lifts, denoted by ${\tilde{\tau }}$ and ${\tilde{\sigma }}$, to ${\tilde{G}}^{\mathbb {C}}$. The fixed point group ${\tilde{K}}^{\mathbb {C}}$ of ${\tilde{\sigma }}$ is connected and projects onto $(K^{\mathbb {C}})^0$. The fixed point group of ${\tilde{\tau }}$ in ${\tilde{G}}^{\mathbb {C}}$ is generally not connected, like in the Willmore surface case, where the real elements ${Fix}^{\tau }({\tilde{G}}^{\mathbb {C}}) = Spin(1,n+3)$ of ${\tilde{G}}^{\mathbb {C}}= Spin(1,n+3, {\mathbb {C}})$ form a non-connected group. But it suffices to consider its connected component $({Fix}^{\tau }({\tilde{G}}^{\mathbb {C}}) )^0 = {\tilde{G}}$ which projects onto G under ${\tilde{\pi }}: {\tilde{G}} \rightarrow G$.

From here on we will write, for convenience, $\Lambda G^0_\sigma$ for $(\Lambda G_\sigma )^0$. Then we trivially obtain the disjoint union

$$\begin{aligned} \Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma } = \bigcup \Lambda {\tilde{G}}_{\sigma }\cdot {\tilde{\delta }} \cdot \Lambda ^{+} {\tilde{G}}^{{\mathbb {C}}}_{\sigma }, \end{aligned}$$

(5.4)

where the ${\tilde{\delta }}$’s simply parameterize the different double cosets. Note that in this equation all groups are connected. We can (and will) assume that ${\tilde{\delta }} = e$ occurs. For the corresponding double coset, since the corresponding Lie algebras add to give the full loop algebra, we obtain:

Theorem 5.3

The multiplication $\Lambda {\tilde{G}}_{\sigma }\times \Lambda ^{+} {\tilde{G}}^{{\mathbb {C}}}_{\sigma }\rightarrow \Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$ is a real analytic map onto the connected open subset $\Lambda {\tilde{G}}_{\sigma } \cdot \Lambda ^{+} {\tilde{G}}^{{\mathbb {C}}}_{\sigma } = {\mathcal {I}}^{{\mathcal {U}}}_e \subset \Lambda {\tilde{G}}^{{\mathbb {C}}}_{\sigma }$.

From this the Iwasawa Decomposition Theorem 3.3 for $(\Lambda G^{\mathbb {C}}_\sigma )^0$ follows after an application of the natural projection as above.

Remark 5.4

• (1) We have seen above that the Iwasawa cell with middle term I is open. But also the Iwasawa cell with middle term $\delta _0 = diag(-1,1,1,1,-1,\ldots ,1)$ is open. To verify this we consider $\delta _0^{-1} \Lambda so(1, n+3)_\sigma \delta _0 \oplus \Lambda ^+so(1,n+3,{\mathbb {C}})$ and observe that the first summand is equal to $\Lambda so(1, n+3)$. As a consequence,

  $$\begin{aligned} \delta _0^{-1} \Lambda SO^+(1, n+3)^0_\sigma \delta _0 \cdot \Lambda _{\mathcal {C}}^+SO(1,n+3,{\mathbb {C}})_\sigma \end{aligned}$$

  is open and the claim follows.

• (2) Using work of Peter Kellersch [26] it seems to be possible to show that there are exactly two open Iwasawa cells in this case. We will not need such a statement in this paper.

1.3 On a complementary solvable subgroup of $SO^+(1,3) \times SO(n)$ in $SO(1,n+3,{\mathbb {C}})$

We consider $K^{{\mathbb {C}}}$, the connected complex subgroup of $SO(1,n+3,{\mathbb {C}})$ with Lie algebra $\mathfrak {so}(1,3,{\mathbb {C}})\times \mathfrak {so}(n,{\mathbb {C}})$ considered as Lie algebra of matrices acting on ${\mathbb {C}}^4\oplus {\mathbb {C}}^{n}$. (Hence we consider the “basic” representations of these Lie algebras as introduced in Sect. 2). Clearly $K^{{\mathbb {C}}}=SO(1,3,{\mathbb {C}})\times SO(n,{\mathbb {C}}).$

Theorem 5.5

There exist connected solvable subgroups $S_1 \subset SO^+(1,3,{\mathbb {C}})$ and $S_2 \subset SO(n,{\mathbb {C}})$ such that

$$\begin{aligned} \left( SO^+(1,3) \times SO(n)\right) \times (S_1 \times S_2) \rightarrow \left( SO^+(1,3) \cdot S_1\right) \times \left( SO(n) \cdot S_2\right) \end{aligned}$$

(5.5)

is a real analytic diffeomorphism onto an open subset of $K^{\mathbb {C}}$.

Proof

Since SO(n) is a connected maximal compact subgroup of $SO(n,{\mathbb {C}})$, in $SO(n,{\mathbb {C}})$ we have the classical Iwasawa decomposition $SO(n,{\mathbb {C}})=SO(n)\cdot B,$ where B is a solvable subgroup of $SO(n,{\mathbb {C}})$ satisfying $SO(n)\cap B=\{I\}$.

It thus suffices to consider $SO(1,3,{\mathbb {C}})$ and to prove the existence of a (connected solvable) subgroup $S_1$ of $SO(1,3,{\mathbb {C}})$ such that

$$\begin{aligned} {\mathcal {S}}: SO^+(1,3)\times S_1 \rightarrow SO^+(1,3)\cdot S_1 \end{aligned}$$

(5.6)

is a real analytic diffeomorphism and $SO^+(1,3)\cdot S_1$ is open in $SO(1,3,{\mathbb {C}})$. Note, since the map ${\mathcal {S}}$ is clearly analytic and surjective, it suffices, as we will see below, to verify that it is also open and that $SO^+(1,3)\cap S_1=\{I\}$ holds.

At any rate, we need to find a solvable Lie subalgebra ${\mathfrak {s}}_1$ of $\mathfrak {so}(1,3,{\mathbb {C}})$ satisfying

$$\begin{aligned} \mathfrak {so}(1,3) + {\mathfrak {s}}_1=\mathfrak {so}(1,3,{\mathbb {C}}),\ \ \mathfrak {so}(1,3)\cap {\mathfrak {s}}_1=\{0\}. \end{aligned}$$

(5.7)

Set

$$\begin{aligned} {\mathfrak {s}}_1=\left\{ \left. \left( \begin{array}{cccc} 0 &{} i a_{12} &{} a_{13} &{} ia_{13} \\ ia_{12} &{} 0 &{} a_{23} &{} ia_{23} \\ a_{13} &{} -a_{23} &{} 0 &{} ia_{34} \\ i a_{13} &{} -ia_{23} &{} -ia_{34} &{} 0 \\ \end{array} \right) \right| \ a_{12},a_{34}\in {\mathbb {R}}, a_{13}, a_{23}\in {\mathbb {C}}\right\} . \end{aligned}$$

We see that $\mathfrak {so}(1,3) \cap {\mathfrak {s}}_1=\{0\}$ and $\mathfrak {so}(1,3)^{{\mathbb {C}}}=\mathfrak {so}(1,3)\oplus {\mathfrak {s}}_1$ hold. It is straightforward to see that ${\mathfrak {s}}_1$ is a solvable Lie algebra. Let $S_1$ be the connected Lie subgroup of $SO(1,3,{\mathbb {C}})$ with Lie algebra $Lie(S_1)={\mathfrak {s}}_1$. So we have that the map ${\mathcal {S}}$ is a local diffeomorphism near the identity element by Chapter II, Lemma 2.4 of [23]. This also implies that the map ${\mathcal {S}}$ is open.

Next we finally show that $SO^+(1,3)\cap S_1=I$ holds. We recall that the exponential map $\exp :\mathfrak {so}(1,3)\rightarrow SO^+(1,3)$ is surjective. Then every element of $SO^+(1,3)\cdot S_1$ has the form $\exp ( {\mathfrak {A}}) \exp ( {\mathfrak {B}})\exp ({\mathfrak {C}}),$ with ${\mathfrak {A}}\in \mathfrak {so}(1,3)$, ${\mathfrak {B}}$ contained in the abelian subalgebra of the $2\times 2-$ diagonal blocks in ${\mathfrak {s}}_1$, and ${\mathfrak {C}}$ in the nilpotent subalgebra of ${\mathfrak {s}}_1$ consisting of the “off-diagonal” blocks (Note that for every off-diagonal block Q in ${\mathfrak {s}}_1$ we have $Q^2 = 0$). Let $\exp ( {\mathfrak {A}}) \exp ( {\mathfrak {B}})\exp ({\mathfrak {C}})\in SO^+(1,3)\cap S_1$. Then $\exp ( {\mathfrak {B}})\exp ({\mathfrak {C}})= \exp (\overline{ {\mathfrak {B}}})\exp (\overline{{\mathfrak {C}}})$ and

$$\begin{aligned} \exp (\overline{ {\mathfrak {B}}})^{-1}\exp ( {\mathfrak {B}})= \exp (\overline{{\mathfrak {C}}})\exp ({\mathfrak {C}})^{-1} \end{aligned}$$

follows. As a consequence, $\exp (\overline{ {\mathfrak {B}}})^{-1}\exp ( {\mathfrak {B}})= \exp (\overline{{\mathfrak {C}}})\exp ({\mathfrak {C}})^{-1}=I_4$, i.e. $\exp ( {\mathfrak {B}})=\exp (\overline{ {\mathfrak {B}}})$ and $\exp ({\mathfrak {C}})=\exp (\overline{{\mathfrak {C}}})$. The definition of ${\mathfrak {s}}_1$ now implies $\exp ({\mathfrak {B}})=\exp ({\mathfrak {C}})=I_4$.

To see that the inverse map is real analytic we take a small neighbourhood in $SO^+(1,3) \cdot S_1$ of the form gVs, where V is a small neighbourhood of the identity I. Since locally near I our map is a real analytic diffeomorphism, the claim follows. $\square$

Remark 5.6

We point out that the set $SO^+(1,3)S_1$ is not all of $SO(1,3,{\mathbb {C}})$. For example

$$\begin{aligned} \left( \begin{array}{cccc} \frac{\sqrt{2}}{2} &{} 0 &{} \frac{i\sqrt{2}}{2} &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ \frac{i\sqrt{2}}{2} &{} 0 &{}\frac{\sqrt{2}}{2} &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ \end{array} \right) \end{aligned}$$

is an element of $SO^+(1,3,{\mathbb {C}})$ which is not contained in $SO^+(1,3)S_1$.