Summary
In this article we construct embedded minimal surfaces which are, at least heuristically, derived from Scherk's first and second surface. Our examples are either parametrized by punctured spheres and then have one translational period or one screw motion period; or they are parametrized by rectangular tori and then have one or two translational periods. The helicoidal examples contain nonisometric ∈-deformations in the sense of Rosenberg [R].
Similar content being viewed by others
References
H.B. Lawson, Complete Minimal Surfaces in S3, Ann. of Math. 92 (1970), 335–374
H. Jenkins, J. Serrin, Variational Problems of Minimal Surface Type II, Arch. Rat. Mech. Analysis 21 (1966), 321–342
D. Hoffman, W.H. Meeks III, Complete Embedded Minimal Surfaces of Finite Total Curvature, Bull. A.M.S., 1985, 134–136.
See also: D. Hoffman, New Embedded Minimal Surfaces, Math. Intelligencer 9, No. 3 (1987), 8–21
W.H. Meeks III, H. Rosenberg, The Global Theory of Doubly Periodic Minimal Surfaces. Preprint Paris 1987
R. Osserman, Global Properties of Minimal Surfaces in E3 and En, Ann. of Math. 80 (1964), 340–364
H. Rosenberg, Deformations of Complete Minimal Surfaces, Trans A.M.S. Vol. 295 (1986), 475–489 and with E. Toubiana pp. 491–499
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Karcher, H. Embedded minimal surfaces derived from Scherk's examples. Manuscripta Math 62, 83–114 (1988). https://doi.org/10.1007/BF01258269
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01258269