ON ALGEBRAIC MINIMAL SURFACES IN ℝ³ DERIVING FROM CHARGE 2 MONOPOLE SPECTRAL CURVES

Abstract

We give formulae for minimal surfaces in ℝ3 deriving, via classical osculation duality, from elliptic curves in a line bundle over ℙ1. Specialising to the case of charge 2 monopole spectral curves we find that the distribution of Gaussian curvature on the auxiliary minimal surface reflects the monopole's structure. This is elucidated by the behaviour of the surface's Gauss map.

Lagrangian Curves on Spectral Curves of Monopoles

“…One alternative approach has been to consider the minimal surface in E 3 generated by the Weierstrass representation applied to the spectral curve [6]. This has been carried out in detail for the charge 2 case, where the resulting geometry has been found to be rich and complicated [12]. The set of points in E 3 where the charge 2 spectral curve lines are orthogonal has also been considered in [7], although no explicit calculations were given.…”

Formulae for null curves deriving from elliptic curves

“…If ψ : S * −→ C 3 is given by osculation then it is null. Let ζ be an affine coordinate on P 1 , giving the local coordinates (ζ, η) −→ (ζ, ηd/dζ ) on T. If the affine part of a curve S in T is described in these coordinates by a pair of meromorphic functions (g, f ) on a Riemann surface M, then, with respect to a certain choice of basis for H 0 (P 1 , O(T)) [11], the coordinate functions of the auxiliary null curve ψ : M * −→ C 3 are given by the Weierstrass formulae (1)-(3). Now, Q 0 = C(Q 1 ) fixes a point in P 9 , the space parameterising all quadrics in P 3 .…”

“…In [11], formulae for the null curves that derive, via (1)-(3), from these elliptic curves are given. It turns out that their surprisingly simple form is not a result of the special nature of spectral curves; similar formulae hold in general, for double covers of the complex projective line P 1 , that are branched over four distinct points.…”