３次元双曲型空間の平坦曲面について

國分雅敏^{∗}

東京電機大学工学部

This talk reports on joint work with Masaaki Umehara and Kotaro Yamada.

It is well known that a minimal surface in Euclidean 3-space 𝔼^{3} is locally given by the real part of a null curve in complex Euclidean 3-space ℂ^{3}. Bryant [B] proved that a surface of constant mean curvature one in hyperbolic 3-space ℍ^{3} is locally the projection from the special linear group SL(2, ℂ) of a null curve to ℍ^{3} = SL(2, ℂ)/SU(2). Any null curve in SL(2, ℂ) is described in terms of two meromorphic functions which are called hyperbolic Gauss map and secondary Gauss map. (cf. [UY], [S])

On the other hand, Gálvez, Martínez and Milán [GMM] proved that a flat surface in ℍ^{3} is locally the projection from SL(2, ℂ) of a Legendre curve to ℍ^{3} = SL(2, ℂ)/SU(2).

In this talk, we show that a Legendre curve in SL(2, ℂ) is also described in terms of two meromorphic functions, and discuss some of its application to flat surfaces in ℍ^{3}.

References

[B] R. Bryant, Surfaces of constant mean curvature one in hyperbolic space, Astérisque 154–155, (1987), 321–347.

[GMM] J. A. Gálvez, A. Martínez and F. Milán, Flat surfaces in the hyperbolic 3-space, Math. Ann. 316 (2000), 419–435.

[S] A. J. Small, Surfaces of Constant Mean Curvature 1 in H^{3} and Algebraic Curves on a Quadric, Proc. Amer. Math. Soc. 122 (1994), 1211– 1220.

[UY] M. Umehara and K. Yamada, Surfaces of constant mean curvature-c in H^{3}(−c^{2}) with prescribed hyperbolic Gauss map, Math. Ann. 304 (1996), 203–224.

∗kokubu@cck.dendai.ac.jp