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Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature. II. (English) Zbl 1058.53008
This paper is a continuation of a preceding work of the same authors with the same title. In the present paper they classify all complete surfaces of constant mean curvature 1 in hyperbolic 3-space $$H^3$$ which have total absolute curvature not exceeding $$4\pi$$. As the authors point out, the total curvature of such surfaces need not be an integer multiple of $$4\pi$$. The classification given is explicit in terms of the holomorphic data of the surface, namely the secondary Gauss map and the Hopf differential. Here the secondary Gauss map of a constant mean curvature 1 surface $$f: M\to H^3$$ is a local isometry from $$(\widetilde M, -Kds^2)$$ into $$\mathbb{C}\mathbb{P}^1$$ endowed with the Fubini-Study metric, where $$\widetilde M$$ denotes the universal cover of $$M$$ and $$ds^2$$ and $$K$$ are the induced metric and the Gauss curvature of $$f$$, respectively.
For part I see W. Rossman, M. Umehara and K. Yamada [Hiroshima Math. J. 34, No. 1, 21–56 (2004; Zbl 1088.53004)].

##### MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 53A35 Non-Euclidean differential geometry 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Zbl 1088.53004
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##### References:
 [1] J. L. M. Barbosa and A. G. Colares, Minimal Surfaces in $$\R^3$$, Lecture Notes in Math. 1195, Springer-Verlag, Berlin, 1986. · Zbl 0609.53002 [2] L. Bieberbach, Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt, Zweite Auflage, Springer-Verlag, Berlin-New York, 1965. · Zbl 0124.04603 [3] R. Bryant, Surfaces of mean curvature one in hyperbolic space, Astérisque 154–155 (1987), 321–347. · Zbl 0635.53047 [4] W. Chen and C. Li, What kinds of singular surfaces can admit constant curvature?, Duke Math. J. 78 (1995), 437–451. · Zbl 0854.53036 [5] C. C. Chen and F. Gackstatter, Elliptische und hyperelliptische Funktionen und vollständige Minimalflächen vom Enneperschen Typ, Math. Ann. 259 (1982), 359–369. · Zbl 0468.53008 [6] P. Collin, L. Hauswirth and H. Rosenberg, The geometry of finite topology Bryant surfaces, Ann. of Math. (2) 153 (2001), 623–659. · Zbl 1066.53019 [7] R. Sa Earp and E. Toubiana, On the geometry of constant mean curvature one surfaces in hyperbolic space, Illinois J. Math. 45 (2001), 371–401. · Zbl 0997.53042 [8] R. Sa Earp and E. Toubiana, Meromorphic data for mean curvature one surfaces in hyperbolic space, · Zbl 1111.53048 [9] F. J. Lopez, The classification of complete minimal surfaces with total curvature greater than $$-12\pi$$, Trans. Amer. Math. Soc. 334 (1992), 49–74. · Zbl 0771.53005 [10] R. Osserman, A Survey of Minimal Surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. · Zbl 0209.52901 [11] W. Rossman and K. Sato, Constant mean curvature surfaces with two ends in hyperbolic space, Experimental Math. 7(2) (1998), 101–119. · Zbl 0980.53081 [12] W. Rossman, M. Umehara and K. Yamada, Irreducible constant mean curvature $$1$$ surfaces in hyperbolic space with positive genus, Tôhoku Math. J. 49 (1997), 449–484. · Zbl 0913.53025 [13] W. Rossman, M. Umehara and K. Yamada, A new flux for mean curvature $$1$$ surfaces in hyperbolic $$3$$-space, and applications, Proc. Amer. Math. Soc. 127 (1999), 2147–2154. · Zbl 0921.53030 [14] W. Rossman, M. Umehara and K. Yamada, Mean curvature $$1$$ surfaces in hyperbolic $$3$$-space with low total curvature I, preprint, math.DG/0008015. · Zbl 1088.53004 [15] W. Rossman, M. Umehara and K. Yamada, Period problems for mean curvature one surfaces in $$H^3$$ (with application to surfaces of low total curvature), to appear in “Surveys on Geometry and Integrable Systems”, Advanced Studies in Pure Mathematics, Mathematical Society of Japan. · Zbl 1171.53041 [16] 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. · Zbl 0823.53044 [17] M. Troyanov, Metric of constant curvature on a sphere with two conical singularities, Differential Geometry (Peñiscda, 1988), 296–306, Lecture Notes in Math. 1410, Springer-Verlag, 1989. · Zbl 0697.53037 [18] M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), 793–821. · Zbl 0724.53023 [19] M. Umehara and K. Yamada, Complete surfaces of constant mean curvature-$$1$$ in the hyperbolic $$3$$-space, Ann. of Math. 137 (1993), 611–638. JSTOR: · Zbl 0795.53006 [20] M. Umehara and K. Yamada, A parameterization of Weierstrass formulae and perturbation of some complete minimal surfaces of $$\R^3$$ into the hyperbolic $$3$$-space, J. Reine Angew. Math. 432 (1992), 93–116. · Zbl 0757.53033 [21] 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. · Zbl 0841.53050 [22] M. Umehara and K. Yamada, Another construction of a CMC-$$1$$ surface in $$H^3$$, Kyungpook Math. J. 35 (1996), 831–849. · Zbl 1004.53043 [23] M. Umehara and K. Yamada, A duality on CMC-$$1$$ surface in the hyperbolic $$3$$-space and a hyperbolic analogue of the Osserman Inequality, Tsukuba J. Math. 21 (1997), 229–237. · Zbl 1027.53010 [24] M. Umehara and K. Yamada, Metrics of constant curvature one with three conical singularities on the $$2$$-sphere, Illinois J. Math. 44 (2000), 72–94. · Zbl 0958.30029 [25] Z. Yu, Value distribution of hyperbolic Gauss maps, Proc. Amer. Math. Soc. 125 (1997), 2997–3001. · Zbl 0903.53004 [26] Z. Yu, The inverse surface and the Osserman Inequality, Tsukuba J. Math. 22 (1998), 575–588. · Zbl 0973.53509
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