Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

How many maximal surfaces do correspond to one minimal surface?

HENRIQUE ARAÜJOa1 and MARIA LUIZA LEITEa1

a1 Departamento de Matemática, Universidade Federal de Pernambuco, 50.740-540 Recife, Brazil. e-mail: henrique@dmat.ufpe.br, mll@hotlink.com.br

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.

(Received December 15 2007)

(Revised April 25 2008)