×

Deforming the singly periodic genus-one helicoid. (English) Zbl 1116.53301

Summary: The Weierstrass data are derived – from geometric assumptions – for a family of screw-motion-invariant minimal surfaces asymptotic to the helicoid. The period problem for these data is solved numerically and the the surfaces are approximated using adaptive mesh methods. These simulations give strong evidence that the family exists, is continuous, consists of embedded surfaces, and limits to the genus-one helicoid.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
PDFBibTeX XMLCite
Full Text: DOI Euclid EuDML

References:

[1] Hoffman D., Encyclopedia of Mathematics pp 5– (1997)
[2] Hoffman D., Global Analysis and Modem Mathematics pp 119– (1993)
[3] Hoffman D., Commentarii Mathematici Helvetici 74 pp 248– (1999) · Zbl 0958.53006 · doi:10.1007/s000140050088
[4] Traizet, M. ”Construction of minimal surfaces by gluing weierstrass representations.”. Proceedings of the Clay Mathematics Institute 2001 Summer School on the Global Theory of Minimal Surfaces. Edited by: Hoffman, D. [Traizet 01] · Zbl 1103.53004
[5] Traizet M., Journal of the Inst. Math. Jussieu 1 (1) pp 145– (2002)
[6] Traizet M., ”An embedded minimal surface with no symmetries.” (2002) · Zbl 1054.53014
[7] Traizet, M. and Weber, M. 2002. [Traizet and Weber 02], In preparation
[8] Weber W., On the embeddedness of the genus one helicoid. (2000)
[9] Weber M., J. Geom. Analysis 12 (2) (2001)
[10] Weber M., ”An embedded genus-one helicoid constructed as the limit of periodic embedded minimal surfaces.”
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.