A spectral curve approach to Lawson symmetric CMC surfaces of genus 2

Abstract

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat $SL(2,{\mathbb {C}})$-connections. In this the paper we parametrize the moduli space of flat $SL(2,{\mathbb {C}})$-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson’s geometric construction. The parametrization uses Hitchin’s abelianization procedure to write such connections explicitly in terms of flat line bundles on a complex 1-dimensional torus. This description is used to develop a spectral curve theory for the Lawson surface. This theory applies as well to other CMC and minimal surfaces with the same holomorphic symmetries as the Lawson surface but different Riemann surface structure. Additionally, we study the space of isospectral deformations of compact minimal surface of genus $g\ge 2$ and prove that it is generated by simple factor dressing.

Appendices

Appendix A: The associated family of flat connections

In this appendix we briefly recall the gauge theoretic description of minimal surfaces in $S^3$ which is due to Hitchin [15]. For more details, one can also consult [18].

The Levi-Civita connection of the round $S^3$ is given with respect to the left trivialization $TS^3=S^3\times \mathfrak {I}{\mathbb {H}}$ as

$$\begin{aligned} \nabla =d+\frac{1}{2}\omega , \end{aligned}$$

where $\omega$ is the Maurer–Cartan form of $S^3$ which acts via adjoint representation.

The hermitian complex rank 2 bundle $V=S^3\times {\mathbb {H}}$ with complex structure given by right multiplication with $i\in {\mathbb {H}}$ is a spin bundle for $S^3:$ The Clifford multiplication is given by $TS^3\times V\rightarrow V; (\lambda , v)\mapsto \lambda v$ where $\lambda \in \mathfrak {I}{\mathbb {H}}$ and $v\in {\mathbb {H}}$, and this identifies $TS^3$ as the skew symmetric trace-free complex linear endomorphisms of $V$. There is a unique complex unitary connection on $V$ which induces on $TS^3\subset End(V)$ the Levi-Civita connection. It is given by

$$\begin{aligned} \nabla =\nabla ^{spin}=d+\frac{1}{2}\omega , \end{aligned}$$

where the $\mathfrak {I}{\mathbb {H}}$-valued Maurer-Cartan form acts by left multiplication in the quaternions.

Let $M$ be a Riemann surface and $f:M\rightarrow S^3$ be a conformal immersion. Then the pullback $\phi =f^*\omega$ of the Maurer-Cartan form satisfies the structural equations

$$\begin{aligned} d^\nabla \phi =0, \end{aligned}$$

(22)

where $\nabla =f^*\nabla =d+\frac{1}{2}\phi$. The conformal map $f$ is minimal if and only if it is harmonic, i.e., if

$$\begin{aligned} d^\nabla *\phi =0. \end{aligned}$$

(23)

holds. Let

$$\begin{aligned} \frac{1}{2}\phi =\varPhi -\varPhi ^* \end{aligned}$$

be the decomposition of $\phi \in \varOmega ^1(M;f^*TS^3)\subset \varOmega ^1(M;End_0(V))$ into the complex linear and complex anti-linear parts. As $f$ is conformal

$$\begin{aligned} \det \varPhi =0. \end{aligned}$$

Note that $f$ is an immersion if and only if $\varPhi$ is nowhere vanishing. In that case $\ker \varPhi =S^*$ is the dual to the holomorphic spin bundle $S$ associated to the immersion. The Eqs. 22 and 23 are equivalent to

$$\begin{aligned} \nabla ''\varPhi =0, \end{aligned}$$

(24)

where $\nabla ''=\frac{1}{2}(d^\nabla +i*d^\nabla )$ is the underlying holomorphic structure of the pull-back of the spin connection on $V$. Of course (24) does not contain the property that $\nabla -\frac{1}{2}\phi =d$ is trivial. Locally this is equivalent to

$$\begin{aligned} F^\nabla =[\varPhi \wedge \varPhi ^*] \end{aligned}$$

(25)

as one easily computes.

From (24) and 25 one sees that the associated family of connections

$$\begin{aligned} \nabla ^\lambda :=\nabla +\lambda ^{-1}\varPhi -\lambda \varPhi ^* \end{aligned}$$

(26)

is flat for all $\lambda \in {\mathbb {C}}^*$, unitary along $S^1\subset {\mathbb {C}}^*$ and trivial for $\lambda =\pm 1$. This family contains all the informations about the surface, i.e., given such a family of flat connections one can reconstruct the surface as the gauge between $\nabla ^1$ and $\nabla ^{-1}$. Using Sym–Bobenko formulas one can also make CMC surfaces in $S^3$ and ${\mathbb {R}}^3$ out of the family of flat connections. These CMC surfaces do not close in general.

The family of flat connections can be written down in terms of the well-known geometric data associated to a minimal surface:

Proposition 7

Let $f:M\rightarrow S^3$ be a conformal minimal immersion with associated complex unitary rank 2 bundle $(V,\nabla )$ and with induced spin bundle $S$. Let $V= S^{-1}\oplus S$ be the unitary decomposition, where $S^{-1}=\ker \varPhi \subset V$ and $\varPhi$ is the $K$-part of the differential of $f$. The Higgs field $\varPhi \in H^0(M,KEnd_0(V))$ can be identified with

$$\begin{aligned} \varPhi =\frac{1}{2} \in H^0(M;KHom(S,S^{-1})), \end{aligned}$$

and its adjoint $\varPhi ^*$ is given by $ivol$ where $vol$ is the volume form of the induced Riemannian metric. The family of flat connections is given by

$$\begin{aligned} \nabla ^\lambda ={\left( \begin{array}{c@{\quad }c}\nabla ^{spin^*} &{} -\frac{i}{2} Q^*\\ -\frac{i}{2} Q &{} \nabla ^{spin}\end{array}\right) }+\lambda ^{-1}\varPhi -\lambda \varPhi ^*, \end{aligned}$$

where $\nabla ^{spin}$ is the spin connection corresponding to the Levi-Civita connection on $M$ and $Q$ is the Hopf differential of $f$.

Appendix B: Lawson’s genus 2 surface

We recall the construction of Lawson’s minimal surfaces of genus 2 in $S^3$, see [21]. Consider the round 3-sphere

$$\begin{aligned} S^3=\{(z,w)\in {\mathbb {C}}^2\mid |z|^2+|w|^2=1\}\subset {\mathbb {C}}\oplus {\mathbb {C}}\end{aligned}$$

and the geodesic circles $C_1=S^3\cap ({\mathbb {C}}\oplus \{0\})$ and $C_2=S^3\cap (\{0\}\oplus {\mathbb {C}})$ on it. Take the six points

$$\begin{aligned} Q_k=(e^{i\frac{\pi }{3}(k-1)},0)\in C_1 \end{aligned}$$

in equidistance on $C_1$, and the four points

$$\begin{aligned} P_k=(0,e^{i\frac{\pi }{2}(k-1)})\in C_2 \end{aligned}$$

in equidistance on $C_2$. A fundamental piece of the Lawson surface is the solution to the Plateau problem for the closed geodesic convex polygon $\varGamma =P_1Q_2P_2Q_1$ in $S^3$. This means that it is the smooth minimal surface which is area minimizing under all surfaces with boundary $\varGamma$. To obtain the Lawson surface one reflects the fundamental piece along the geodesic through $P_1$ and $Q_1$, then one rotates everything around the geodesic $C_2$ by $\frac{2}{3}\pi$ two times, and in the end one reflects the resulting surface across the geodesic $C_1$. Lawson has shown that the surface obtained in this way is smooth at all points. It is embedded, orientable and has genus 2. The umbilics, i.e., the zeros of the Hopf differential $Q$ are exactly at the points $P_1,..,P_4$ of order 1.

A generating system of the symmetry group of the Lawson surface is given by

• the ${\mathbb {Z}}^2$-action generated by $\varphi _2$ with $(a,b)\mapsto (a,-b);$ it is orientation preserving on the surface and its fixed points are $Q_1,..Q_6;$

• the ${\mathbb {Z}}_3$-action generated by the rotation $\varphi _3$ around $P_1P_2$ by $\frac{2}{3}\pi$, i.e., $(a,b)\mapsto (e^{i\frac{2}{3}\pi }a,b)$, which is holomorphic on $M$ with fixed points $P_1,..,P_4;$

• the reflection at $P_1Q_1$, which is antiholomorphic: $\gamma _{P_1Q_1}(a,b)=(\bar{a},\bar{b});$

• the reflection at the sphere $S_1$ corresponding to the real hyperplane spanned by $(0,1)$, $(0,i)$ and $(e^{\frac{1}{6}\pi i},0)$, with $\gamma _{S_1}(a,b)=(e^{\frac{\pi }{3}i}\bar{a}, b);$ it is antiholomorphic on the surface,

• the reflection at the sphere $S_2$ corresponding to the real hyperplane spanned by $(1,0)$, $(i,0)$ and $(0,e^{\frac{1}{4}\pi i})$, which is antiholomorphic on the surface and satisfies $\gamma _{S_2}(a,b)=(a,i\bar{b})$.

Note that all these actions commute with the ${\mathbb {Z}}_2$-action. The last two fix the polygon $\varGamma$. They and the first two map the oriented normal to itself. The third one maps the oriented normal to its negative.

Using the symmetries, one can determine the Riemann surface structure of the Lawson surface $f:M\rightarrow S^3$ as well as the other holomorphic data associated to it:

Proposition 8

The Riemann surface $M$ associated to the Lawson genus 2 surface is the threefold covering $\pi :M\rightarrow {\mathbb {CP}}^1$ of the Riemann sphere with branch points of order 2 over $\pm 1,\pm i\in {\mathbb {CP}}^1$, i.e., the compactification of the algebraic curve

$$\begin{aligned} y^3=\frac{z^2-1}{z^2+1}. \end{aligned}$$

The hyper-elliptic involution is given by $(y,z)\mapsto (y,-z)$ and the Weierstrass points are $Q_1,..,Q_6$. The Hopf differential of the Lawson genus 2 surface is given by

$$\begin{aligned} Q=\pi ^*\frac{ir}{z^4-1}(dz)^2 \end{aligned}$$

for a nonzero real constant $r\in {\mathbb {R}}$ and the spin bundle $S$ of the immersion is

$$\begin{aligned} S=L(Q_1+Q_3-Q_5). \end{aligned}$$