DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 203–206

CONSTRUCTION OF CMC–SURFACES IN S^{3} FORM A PAIR

OF NON–CONFORMAL HARMONIC MAPS INTO S^{2}

R. Miyaoka

Abstract. We report our recent result (joint work with R. Aiyama, K. Akutagawa and M. Umehara) on a global correpondence between CMC surfaces in P^{3}(c^{2}) or in S^{3}(c^{2}) and a pair of non-conformal harmonic maps into S^{2} in a same associated family.

Consider a problem to construct a surface with prescribed Gauss map. For surfaces in R^{3}, the Gauss map is a map into S^{2}, and satisfies some integrability condition. Actually, let M be a Riemann surface with complex parameter z, and g : M → S^{2} be a map satisfying

(1)  H(g_{zz̄} − \frac{2ḡg_{z}gz̄}{1 + |g|^{2}} ) = H_{z}gz̄ ,

for a function H on M, where g is identified with g followed by a stereographic projection to C. Kenmotsu shows that this is necessary and sufficient for the existence of a surface (possibly with singularities) x_{g} : M → R^{3} with the mean curvature H and the Gauss map g, if M is contractible and H ≠ 0 on M. In fact, x_{g} is represented by the so-called Kenmotsu formula [K]

(2)  x_{g} = (∫ ω_{1}, ∫ ω_{2}, ∫ ω_{3}) ,

where

\begin{aligned}ω_{1} & = ℜ\{ψ(1 − g^{2})dz\}, ω_{2} = ℜ\{iψ(1 + g^{2})dz\}, ω_{3} = ℜ\{2ψgdz\}, \\ ψ & = −\frac{ḡz̄}{H(1 + |g|^{2})^{2}} . \end{aligned}

When g is an holomorphic map, (2) corresponds to the Weierstrass formula for minimal surfaces in R^{3}. When g is a non-conformal harmonic map, H reduces to a constant and we obtain a CMC (constant mean curvature) surface by (2).

1991 Mathematics Subject Classification. Primary 53C42, 53A10; Secondary 58E20.

Key words and phrases. CMC surface, Gauss map.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Recently, representation formulas in spin version are obtained, for instance, by Konopelchenko [Ko], who uses the Dirac operator

D = ( \begin{matrix}0 & ∂ \\ −¯∂ & 0 \end{matrix} ) + ( \begin{matrix}U & 0 \\ 0 & U \end{matrix} ) ,

with the half potential U(z, z̄), which is a real valued function (essentially the mean curvature function, see (5)), to describe an integrability condition for the construction of surfaces from a given pair of complex valued functions (ψ_{1}, ψ_{2}), which is essentially the Gauss map. The surface is given by

\begin{aligned}x^{1} & = \frac{i}{2} ∫ ((ψ̄_{2}^{2} + ψ_{1}^{2})dz − (ψ̄_{1}^{2} + ψ_{2}^{2})dārz) , \\ x^{2} & = \frac{1}{2} ∫ ((ψ̄_{2}^{2} − ψ_{1}^{2})dz − (ψ̄_{1}^{2} − ψ_{2}^{2})dz̄) , \\ x^{3} & = − ∫ (ψ_{1}ψ̄_{2}dz + ψ̄_{1}ψ_{2}dz̄) , \end{aligned} (3)

under the integrabiblity condition simply expressed by

(4)  Dψ = 0 .

This corresponds to various equations in integrable systems, according to various properties of the surfaces, such as constantness of the Gaussian curvature, critical points of the Willmore functional, etc. The Gauss map is given by ψ_{1}/ψ̄_{2}, and the metric tensor F (z, z̄)^{2}dzdz̄ and the mean curvature function H are given respectively by

(5)  F (z, z̄) = |ψ_{1}(z, z̄)|^{2} + |ψ_{2}(z, z̄)|^{2}, H = 2U/F .

These expressions imply that we may consider a surface given by (2) or (3) as a surface with prescribed Gauss map as well as a surface with prescribed mean curvature function, because g and H ((ψ_{1}, ψ_{2}) and H, respectively) are related by (1) or (5), if the surface exists.

Aiyama and Akutagawa consider a similar problem in S^{3}(c^{2}) and H^{3}(−c^{2}), where ±c^{2} denote the curvature, and obtain representation formulas for surfaces with prescribed mean curvature function H [AA1,2]. We restrict our argument to a surface in S^{3}(c^{2}) in this article. Here, the Gauss map means the generalized Gauss map given by G : M → Gr^{+}(2, 2) ≅ S^{2}(1) × S^{2}(1), hence consists of a pair of maps g_{i} : M → S^{2}, G = (g_{1}, g_{2}). In [AA2], they clarify the mechanism how to construct a surface in S^{3}(c^{2}) from a given ν = g_{2} (followed by a stereographic projection) satisfying

ν_{zz̄} − \frac{2ν̄}{1 + |ν|^{2}} ν_{z}νz̄ = \frac{1}{H + ic}H_{z}νz̄ ,

where H is supposed to be the mean curvature function. (Compare with (1)).

Here we recall that when H is constant, g_{1} and g_{2} are non-conformal harmonic maps in the same associated family [HO, S]. In fact, g_{2} is related to g_{1} by g_{2} = g_{1}^{α}

A GLOBAL CORRESPONDENCE BETWEEN CMC–SURFACES … 205

where g_{2} has the Hopf differential multiplied by e^{2iα} to that of g_{1}, and cos α = \sqrt{H^{2}/(H^{2} + c^{2})} [AA2, M]. In other words, from a non-conformal harmonic map g : M → S^{2}, the partner g^{α} generating a generalized Gauss map of a CMC-H surface in S^{3}(c^{2}), is determined uniquely by cos α = \sqrt{H^{2}/(H^{2} + c^{2})}. Up to now, we consider a local correpondence.

In [AAMU], we obtain a global correspondence, which is a direct consequence of the representation formula in [AA2, Theorem 4.2], but can be also proved without this formula:

Theorem. Let g : M → S^{2} be a non-conformal harmonic map. Suppose that there exists a real number α ∈ (0, π/2] such that the associated harmonic map g^{α} is single-valued on M. Then there exists a branched conformal CMC-H immersion x of M into P^{3}(c^{2}) with the generalized Gauss map G = (g, g^{α}), where cos α = \sqrt{H^{2}/(H^{2} + c^{2})}.

If we take a suitable covering, the argument implies a similar results for CMC surfaces in S^{3}(c^{2}).

As is well known, harmonic maps into symmetric spaces exist in an S^{1} family {g^{α}} consisting of maps from the universal cover of M. When they are noncongruent, it is interesting to know for which spectral parameter α, g^{α} is single valued on M. When M is compact, those α are isolated in S^{1} [AAMU]. From compact CMC surfaces in S^{3}(c^{2}), we obtain a pair of such parameters. An interesting open question is whether such pair is unique or not in S^{1}.

Finally, we remark that Bobenko’s formula [B1, Theorem 14.1] for CMC surfaces in S^{3}(c^{2}), which is obtained in the same context as Sym’s formula, also implies this theorem. Moreover we obtain

Corollary [B2, §5], [AAMU]. Let g : M → S^{2} be a non-conformal harmonic map. Then the branched CMC-immersion x_{g} : M̃ → R^{3} with the Gauss map g given by

the Kenmotsu formula [K] is single-valued on M if and only if \frac{dρ_{θ}}{dθ} |_{θ=0} = 0 where

ρ_{θ} : π_{1}(M) → SU(2) is the monodromy representation of g^{θ}.

References

[AA1] Aiyama, R., Akutagawa, K., Kenmotsu type representation formula for surfaces with prescribed mean curvature in ℍ^{3}(−c^{2}) and adjusting it for CMC surfaces, preprint.

[AA2] , Kenmotsu type representation formula for surfaces with prescribed mean curvature in 𝕊^{3}(c^{2}) and adjusting it for CMC surfaces, preprint.

[AAMU] Aiyama, R., Akutagawa, K., Miyaoka, R., Umehara, M., A global correspondence between CMC-surfaces in S^{3} and pairs of non-conformal harmonic maps into S^{2}, preprint.

[B1] Bobenko, A., Constant mean curvarture surfaces and integrable equations, Russian Math. Surveys 46 (1991), 1–45.

[B2] , Surfaces in terms of 2 by 2 matrices: Old and new integrable cases, Harmonic maps and integrable systems (Eds. P. Fordy and J. C. Wood), Vieweg, 1994, pp. 83–127.

[HO] Hoffman Jr., D. A., Osserman, R., On the Gauss map of surfaces in R^{3} and R^{4}, Proc. London Math. Soc. 50 (1985), 27–56.

[K] Kenmotsu, K., Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979), 89-99.

[Ko] Konopelchenko, B. G., Induced surfaces and their integrable dynamics, Studies in Appl. Math. 96 (1996), 9–51.

[M] Miyaoka, R., The splitting and deformations of the generalized Gauss map of compact CMC surfaces, Tôhoku Math. J. 51 (1999), 35–53.

[S] Seaman, W., On surfaces in R^{4}, Proc. Amer. Math. Soc. 94 (1985), 467–470.

Department of Mathematics, Sophia University Kioi-cho, Chiyoda-ku, Tokyo 1028554, JAPAN E-mail: r-miyaok@hoffman.cc.sophia.ac.jp