Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-18T15:59:32.131Z Has data issue: false hasContentIssue false
  • English
  • Français

How many maximal surfaces do correspond to one minimal surface?

Published online by Cambridge University Press:  01 January 2009

HENRIQUE ARAÜJO  and
MARIA LUIZA LEITE
HENRIQUE ARAÜJO
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco, 50.740-540 Recife, Brazil. e-mail: henrique@dmat.ufpe.br, mll@hotlink.com.br
MARIA LUIZA LEITE
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco, 50.740-540 Recife, Brazil. e-mail: henrique@dmat.ufpe.br, mll@hotlink.com.br
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss the minimal-to-maximal correspondence between surfaces and show that, under this correspondence, a congruence class of minimal surfaces in 3 determines an 2-family of congruence classes of maximal surfaces in 3. It is proved that further identifications among these classes may exist, depending upon the subgroup of automorphisms preserving the Hopf differential on the underlying Riemann surface. The space of maximal congruence classes inherits a quotient topology from 2. In the case of the Scherk minimal surface, the subgroup has order two and the quotient space is topologically a disc with boundary. Other classical examples are discussed: for the Enneper minimal surface, one obtains a non-Hausdorff space; for the minimal catenoid, a closed interval.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 165 - 175
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Calabi, E.Examples of Bernstein problems for some nonlinear equations. Proc. Symp. Pure Math. 15 (1970), 223–230.CrossRefGoogle Scholar
[2]Fernández, I., López, F. J. and Souam, R.The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space 3. Math. Ann. 332 (2005), 605–643.CrossRefGoogle Scholar
[3]Kobayashi, O.Maximal surfaces in the 3-dimensional Minkowski space 3. Tokyo J. Math. 6 (1983), 297–309.CrossRefGoogle Scholar
[4]Leite, M. L. Orthogonal systems of cycles in 2 and maximal surfaces in 3 with planar lines of curvature, preprint (2007).Google Scholar
[5]López, F. J., López, R. and Souam, R.Maximal surfaces of Riemann type in Lorentz-Minkowski space 3. Michigan Math. J. 47 (2000), 469–497.CrossRefGoogle Scholar
[6]López, F. J. and Martín, F.Complete minimal surfaces in 3 Pub. Mat. 43, No. 2, (1999), 341–449.CrossRefGoogle Scholar
[7]Osserman, R.A Survey of minimal surfaces, second edition (Dover Publications, 1986).Google Scholar