Quadrics and Scherk towers

S Fujimori,K Yamada,M Kokubu,U Hertrich-Jeromin,M Umehara

doi:10.1007/s00605-017-1075-5

Abstract

We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic functions. The curves of type change for real isothermic surfaces of mixed causal type turn out to be aligned with the real curvature line net.

Highlights

Though quadrics belong to the most thoroughly investigated surfaces there are still mysterious aspects in their geometry
Quadrics can reasonably be studied in a variety of ambient geometries—Möbius or conformal geometry not being one of these geometries since Möbius transformations generally do not map a quadric to another quadric
Motivated by the existence of the 1-parameter family of associated minimal surfaces for a given minimal surface, that are isometric and feature parallel tangent planes, Christoffel classified those surfaces that admit a non-trivial partner surface that is conformally related to the first by parallel tangent planes, cf. [2]: apart from associated minimal surfaces these are precisely the isothermic surfaces, which admit a partner surface that is generically unique up to scaling so that the relation is orientation reversing: we will refer to this partner surface of an isothermic surface as its Christoffel dual

Summary

Introduction

Though quadrics belong to the most thoroughly investigated surfaces there are still mysterious aspects in their geometry. As the notions of isothermic surface and Christoffel duality are Möbius geometric and Euclidean notions, respectively, this feature of the ellipsoid and its Christoffel dual appear to be a highly unlikely coincidence: the principal aim of this paper is to shed further light on this coincidence and to obtain a better understanding of the reasons behind it. As the Enneper– Weierstrass representation for minimal surfaces in Euclidean space provides a method to explicitly determine the Christoffel dual of a tri-axial ellipsoid, so do Kobayashi’s and Konderak’s Weierstrass type representations for maximal and timelike minimal surfaces in Minkowski space to find the Christoffel duals of hyperboloids, see [10] resp [11]. Part of our investigations is independent of the existence of and relation to a zero mean curvature surface: we obtain explicit representations of Christoffel duals in Euclidean as well as Minkowski ambient geometries, see Cor 3.3, Cor 4.2, Cor 4.3, Cor 5.1 and Cor 5.2. Though we restrict ourselves to quadrics in Minkowski space that are aligned to the timelike axis of the ambient space, this restriction is not essential: the same methods will lead to similar results if a surface is in a more general position as the nature of the occurring differential equation will not change—computations and formulas will be less transparent

The ellipsoid

Scherk’s saddle tower

The spacelike hyperboloid

The timelike hyperboloid