A reducibility for a triply periodic minimal surface. (English) Zbl 1453.53010

Shoda, Toshihiro (ed.) et al., Differential geometry and Tanaka theory. Differential system and hypersurface theory. Proceedings of the international conference, RIMS, Kyoto University, Japan, January, 24–28, 2011. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 82, 41-55 (2019).
Summary: In this paper, we define a reducibility for a triply periodic minimal surface, and study hyperelliptic minimal surfaces and trigonal minimal surfaces in terms of the reducibility. Restrictions on the topological type of the above minimal surfaces are given.
For the entire collection see [Zbl 1437.53042].


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization
Full Text: DOI Euclid


[1] E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of Algebraic Curves. Vol. I, Springer-Verlag, 1985. · Zbl 0559.14017
[2] M. Coppens, C. Keem and G. Martens, Primitive linear series on curves, Manuscripta. Math.,77(1992), 237-264. · Zbl 0786.14016
[3] N. Ejiri, A differential-geometric Schottky problem, and minimal surfaces in tori, Contemp. Math.,308(2002), 101-144. · Zbl 1071.58012
[4] D. A. Hoffman and R. Osserman, The geometry of the generalized Gauss map, Mem. Amer. Math. Soc.,28, no. 236. · Zbl 0469.53004
[5] H. Karcher, The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscripta. Math.,64 (1989), no. 3, 291-357. · Zbl 0687.53010
[6] W. H. Meeks III, The theory of triply periodic minimal surfaces, Indiana Univ. Math. J.,39(1990), no. 3, 877-936. · Zbl 0721.53057
[7] T. Nagano and B. Smyth, Minimal varieties and harmonic maps in tori, Comment. Math. Helv.,50(1975), 249-265. · Zbl 0326.53055
[8] T. Nagano and B. Smyth, Periodic minimal surfaces, Comment. Math. Helv.,53(1978), no. 1, 29-55. · Zbl 0384.53025
[9] T. Nagano and B. Smyth, Periodic minimal surfaces and Weyl groups, Acta Math.,145(1980), no. 1-2, 1-27. · Zbl 0449.53042
[10] T.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.