zbMATH — the first resource for mathematics

The structure of singly-periodic minimal surfaces. (English) Zbl 0695.53005
The authors generalize Riemann’s minimal surface by constructing an infinite family of properly embedded periodic minimal surfaces with an infinite number of (flat) ends. The symmetry is a group of translations, but the examples can be “twisted”, so that it becomes screw motion. Conversely it is shown that a properly embedded minimal surface with more than one end and infinite symmetry group either is a catenoid, or it has infinitely many ends, and is invariant under a screw motion.
Reviewer: D.Ferus

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text: DOI EuDML
[1] Callahan, M., Hoffman, D., Meeks III, W.H.: Embedded minimal surface with an infinite number of ends. Invent. Math.96, 459–505 (1989) · Zbl 0676.53004 · doi:10.1007/BF01393694
[2] Choi, T., Meeks III, W.H., White, B.: A rigidity theorem for properly embedded minimal surfaces in \(\mathbb{R}\)3. J. Differ. Geom. · Zbl 0704.53008
[3] Fischer-Colbrie, D.: On complete minimal surfaces with finite Morse index in 3-manifolds. Invent. Math.82, 121–132 (1985) · Zbl 0573.53038 · doi:10.1007/BF01394782
[4] Freedman, M., Hass, J., Scott, P.: Closed geodesics on surfaces. Bull. Lond. Math. Soc.14, 385–391 (1982) · Zbl 0487.53040 · doi:10.1112/blms/14.5.385
[5] Hadamard, J.-J.: Les surfaces à courbures opposées et leurs lignes géodésiques. J. Math. Pures Appl.4, 27–73 (1898) · JFM 29.0522.01
[6] Hardt, R., Simon, L.: Boundary regularity and embedded minimal solutions for the oriented Plateau problem. Ann. Math.110, 439–486 (1979) · Zbl 0457.49029 · doi:10.2307/1971233
[7] Hoffman, D., Meeks III, W.H.: The asymptotic behavior of properly embedded minimal surfaces of finite topology. J. Am. Math. Soc.2, No. 4, 667–681 (1989) · Zbl 0683.53005 · doi:10.1090/S0894-0347-1989-1002088-X
[8] Hoffman, D., Meeks III, W.H.: The strong halfspace theorem for minimal surfaces. Invent. Math. (to appear) · Zbl 0722.53054
[9] Hoffman, D., Meeks III, W.H.: A variational approach to the existence of complete embedded minimal surfaces. Duke J. Math.57, 877–894 (1988) · Zbl 0676.53006 · doi:10.1215/S0012-7094-88-05739-0
[10] Huber, A.: On subharmonic functions and differential geometry in the large. Comment. Math. Helv.32, 181–206 (1957) · Zbl 0080.15001
[11] Karcher, H.: Embedded minimal surfaces derived from Scherk’s examples. Manuscr. Math.62, 83–114 (1988) · Zbl 0658.53006 · doi:10.1007/BF01258269
[12] Karcher, H., Pitts, J.: (personal communications)
[13] Meeks III, W.H., Rosenberg, H.: The geometry of periodic minimal surfaces (Preprint) · Zbl 0807.53049
[14] Meeks III, W.H., Rosenberg, H.: The global theory of doubly periodic minimal surfaces. Invent. Math.97, 351–379 (1989) · Zbl 0676.53068 · doi:10.1007/BF01389046
[15] Meeks III, W.H., Rosenberg, H.: The maximum principle at infinite for minimal surfaces in flat three-manifolds. Comm. Math. Helv. (to appear) · Zbl 0713.53008
[16] Meeks III, W.H., Yau, S.T.: The topological uniqueness theorem of complete minimal surfaces of finite topological type (Preprint) · Zbl 0761.53006
[17] Meeks III, W.H.: Yau, S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z.179, 151–168 (1982) · Zbl 0479.49026 · doi:10.1007/BF01214308
[18] Morse, M.: Collected Papers. World Scientific, Singapore, 1987 · Zbl 0664.01017
[19] Osserman, R.: Minimal surfaces in the large. Comment. Math. Helv.35, 65–76 (1961) · Zbl 0098.34904 · doi:10.1007/BF02567006
[20] Osserman, R.: Global properties of minimal surfaces inE 3 andE n . Ann. Math.80, 340–364 (1964) · Zbl 0134.38502 · doi:10.2307/1970396
[21] Schoen, R.: Uniqueness, symmetry, and embeddedness of minimal surfaces. J. Differ. Geom.18, 791–809 (1983) · Zbl 0575.53037
[22] Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Center for Mathematical Analysis, vol. 3, Canberra, Australia, 1983. Australian National University · Zbl 0546.49019
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.