Josai Mathematical Monographs vol 3 (2001), pp. 37-41

ON SURFACES WITH HOLOMORPHIC GAUSS MAP IN A CERTAIN SYMMETRIC SPACE

MASATOSHI KOKUBU

I would like to talk about some results on analogue of minimal surface theory in a certain symmetric space, which is a joint work [KTUY] with Masaro Takahashi, Masaaki Umehara and Kotaro Yamada.

1. Background

We shall quickly review the fundamental facts about the minimal surface theory in Euclidean 3-space, the CMC-1 surface theory in hyperbolic 3-space and their correspondence.

f: M → R^{3} minimal immersion f: M → H^{3} CMC-1 immersion

Def

mean curvature 0 mean curvature 1 ⟺ ∃ null curve F: M̃ → C^{3} s.t. f = 2 Re F.

∃ null curve

F: M̃ → SL_{2} C s.t. f = F F^{∗}

⟺ f is given by

Re ∫ ( \begin{matrix}1−g^{2} \\ i(1+g^{2}) \\ 2g \end{matrix} ) ω

{f = F F^{∗} where \atop F^{−1}dF = ( \begin{matrix}g & −g^{2} \\ 1 & −g \end{matrix} ) ω}

⟺ Gauss map [∂f]: M → Q_{1} ⊂ P (C^{3}) is holomorphic. hyperbolic Gauss map [∂ff^{−1}]: M → Q_{1} ⊂ P (𝔰𝔩_{2}C) is holomorphic.

1st f.f. (1 + |g|^{2})^{2}ω\overline{ω}  (1 + |g|^{2})^{2}ω\overline{ω}

period ∀[γ] ∈ π_{1}(M), ; ∫_{γ} ( \begin{matrix}1−g^{2} \\ i(1+g^{2}) \\ 2g \end{matrix} ) ω ∈ \sqrt{−1}R^{3} The monodromy group of

dF F^{−1} = ( \begin{matrix}G & −G^{2} \\ 1 & −G \end{matrix} ) Ω

is included in SU_{2}.

M is an oriented surface (with the induced metric), hence a Riemann surface. M̃ denotes the universal cover of M. H^{3} = SL_{2} C/ SU_{2} is embedded in SL_{2} C (known as Cartan embedding).

2. A generalization of the ambient space

Let 𝔤 be a complex semisimple Lie algebra. Then a symmetric space N of noncompact type is associated with 𝔤 as follows:

Let 𝔥 be a compact real form of 𝔤. Let σ_{0} : 𝔤 → 𝔤 denote the complex conjugate with respect to the decomposition 𝔤 = 𝔥 ⊕ \sqrt{−1}𝔥. We denote by G̃ the simplyconnected Lie group with Lie algebra 𝔤 and by σ̃: G̃ → G̃ the involution of G̃ such that (σ̃)_{e} = σ_{0}. Let H̃ be the set of fixed points of σ̃. Then H̃ is a connected Lie subgroup of G̃ with Lie algebra 𝔥.

In these circumstances, we have a Riemannian symmetric pair

(G̃, H̃, σ̃, B|_{\sqrt{−1}𝔥}),

where B is the Killing form of 𝔤. Hence we have a Riemannian symmetric space N = G̃/H̃.

The hyperbolic 3-space is a model of N = G̃/H̃. If we start with 𝔤 = 𝔰𝔩_{2}C, the associated space N is H^{3} = SL_{2} C/ SU_{2}.

Without loss of generality, we may assume that the symmetric space N can be represented as

N(= G̃/H̃) = G/H (G ⊂ SL_{n}C).

(If G̃ is not a subgroup of SL_{n} C, we have only to replace G̃ and H̃ by G = Ad(G̃) and H = Ad(H̃), respectively.)

Moreover we usually consider the Cartan model

N = G/H ⊂ G(⊂ SL_{n}C).

3. Surfaces with holomorphic right Gauss map

Let N = G/H(⊂ G) be a symmetric space defined in the previous section. Let f: M → N be an immersion of an oriented surface. Here we think of f to be a G-valued map. As usual, we regard M as a Riemann surface by the orientation and the induced metric.

Definition 3.1. We call the map ν defined by

ν: p ∈ M ↦ [(f^{∗}μ_{R})^{1,0}|_{p}] ∈ P (𝔤) (3.1)

the right Gauss map. Here, μ_{R} denotes the right-invariant Maurer-Cartan on G and P (𝔤) is the complex projective space of 𝔤.

A surface with holomorphic Gauss map in N is a generalization of a CMC-1 surface in H^{3} from the viewpoint of Gauss map. There are also some similar property at the other point:

1. Bryant type formula. Any immersion f: M → N = G/H of a surface with holomorphic right Gauss map is given by f = F F^{∗} where F: M̃ → G is a null curve.

2. Local correspondence to minimal surfaces. For any surface with holomorphic right Gauss map f: M → N = G/H, there exists a minimal surface in R^{dim G} which is locally isometric to M.

3. Duality, Dual metric. Given a surface with holomorphic right Gauss map f = F F^{∗}, there is another surface with holomorphic right Gauss map defined by f^{#} = (F^{#})(F^{#})^{∗}: M̃ → N = G/H where F^{#} : z ↦ F (z)^{−1}. We call f^{#} the dual of f. Though the dual surface f^{#} is defined on the universal cover M̃ in general, the induced metric ds^{2}^{#} = B(dF F^{−1}, \overline{dF F^{−1}}) is well-defined on M.

The following consideration implies that the study of surfaces with holomorphic Gauss map in N = G/H is not only the analogue but also the generalization of Euclidean minimal surface theory:

N = G/H contains a Euclidean space R^{r} as a totally geodesic submanifold, where r is the rank of the Lie group G. Indeed, the submanifold consisting of diagonal matrices in N = SL_{n}C/ SU_{n}(⊂ SL_{n}C) is isometric to R^{n−1} via the exponential map. It is easy to see that f: M → R^{r} ⊂ N = G/H has holomorphic right Gauss map (as a surface in G/H) if and only if f is a conformal minimal immersion as a surface in R^{r}.

4. Examples

Example 4.1. Let M be a Riemann surface and a, b, c be meromorphic functions on M. We set

F := (\begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}) .

Then F is a null meromorphic map on M^{2}. So we can construct many surfaces having holomorphic right Gauss maps as the projection of such F. For example, if we set M = C \ {0} and a(z) = 1/b(z) = z, c(z) = 1, then it gives a complete surfaces with finite total curvature in SL(3, C)/ SU(3). For a suitable choice of orthonormal basis, one can easily check that the canonical correspondence F_{0} : M̃^{2} → C^{8} is given by

F_{0} := ∫_{z_{0}}^{z}(db − adc, i(db − adc), da, −ida, dc, idc, 0, 0).

F is called a null curve if B(F^{−1}dF, F^{−1}dF) ≡ 0.

Example 4.2. Let M = C \ {0}, and let M̃ be the universal cover of M. Define a holomorphic map F_{μ,a,b}: M̃ → SL(3, C) by

F_{μ,a,b}(z) = [\begin{matrix}\sqrt{\frac{b^{2}+3μ^{2}}{b^{2}−a^{2}}} z^{μ+a} & 0 & \sqrt{\frac{a^{2}+3μ^{2}}{b^{2}−a^{2}}} z^{μ−b} \\ 0 & z^{−2μ} & 0 \\ \sqrt{\frac{a^{2}+3μ^{2}}{b^{2}−a^{2}}} z^{μ+b} & 0 & \sqrt{\frac{b^{2}+3μ^{2}}{b^{2}−a^{2}}} z^{μ−a} \end{matrix}] , (4.1)

where μ, a and b are real constants such that b^{2} > a^{2}. Then F_{μ,a,b} takes values in the 4-dimensional subgroup (C^{∗} × GL(2, C)) ∩ SL(3, C), in particular, it is SL(2, C)- valued if μ = 0.

For F = F_{μ,a,b}, the 1-form α_{F} is computed as follows:

α_{F} = [\begin{matrix}(μ + \frac{ab−3μ^{2}}{a+b})z^{−1} & 0 & −^{\frac{\sqrt{(b^{2}+3μ^{2})(a^{2}+3μ^{2})}}{a+b}}z^{−a−b−1} \\ 0 & −2μz^{−1} & 0 \\ ^{\frac{\sqrt{(b^{2}+3μ^{2})(a^{2}+3μ^{2})}}{a+b}}z^{a+b−1} & 0 & (μ − \frac{ab−3μ^{2}}{a+b})z^{−1} \end{matrix}] dz.

It is verified treat the case when f itself is single-valued on C \ {0}, i.e.,

f = F_{μ,a,b}(F_{μ,a,b})^{∗}, b − a ∈ Z \ {0},

where F_{μ,a,b} is given by (4.1). Then α_{F}^{#} = dF F^{−1} is computed as

α_{F}^{#} = [\begin{matrix}(μ + \frac{ab+3μ^{2}}{b−a} ) z^{−1} & 0 & −(a + b)z^{a−b−1} \\ 0 & −2μz^{−1} & 0 \\ (a + b)z^{b−a−1} & 0 & (μ − \frac{ab+3μ^{2}}{b−a} ) z^{−1} \end{matrix}] dz.

Indeed, α_{F}^{#} is single-valued on C \ {0}. Furthermore, ds^{2}^{#} = tr α^{#}(α^{#})^{∗} is

ds^{2}^{#} = \{\frac{2(a^{2} + 3μ^{2})(b^{2} + 3μ^{2})}{(b − a)^{2}} + (a + b)^{2}(|z|^{2a−2b} + |z|^{2b−2a})\} |z|^{−2} dz d\overline{z}.

If |a − b| = 1, the dual total curvature of f is equal to −4π which satisfies equality in the Chern-Osserman type inequality (see the next section).

5. Total (dual) curvature

We denote by ds^{2} (resp. K) the induced metric (resp. the Gauss curvature) of a surface with holomorphic right Gauss map. Since the metric ds^{2} is locally isometric to the metric of an Euclidean minimal surface, the Gauss curvature K is non-positive. Similarly for the dual metric ds^{2}^{#}, the dual Gauss curvature K^{#} is non-positive.

Theorem 5.1. Let f: M → N = G/H be a complete surface with holomorphic right Gauss map. Then the dual metric is also complete, and the following hold:

\frac{1}{2π} ∫_{M} K^{#}dA^{#} ∈ Z
≤ χ(M) − #{ends},

where χ(M) is the Euler number of M.

It is known that (1/4π) ∫_{M} K^{#}dA^{#} ∈ Z holds in the case of N = H^{3}. However, it fails to hold in general N. (Example 4.2 gives a counterexample.)

On the other hand, the total curvature satisfies neither the integrity nor the Chern-Osserman type inequality.

From the intrinsic condition, we can see that the Cohn-Vossen inequality

\frac{1}{2π} ∫_{M} KdA ≤ χ(M)

holds. Umehara and Yamada proved the inequality

\frac{1}{2π} ∫_{M} KdA < χ(M)

in the case of N = H^{3}. It is still unknown whether there is a surface with holomorphic Gauss map in the other symmetric space N = G/H which satisfies equality in the Cohn-Vossen inequality.

References

[KTUY] M. Kokubu, M. Takahashi, M. Umehara and K. Yamada, An analogue of minimal surface theory in SL(n, C)/ SU(n), preprint

Tokyo Denki University