Lagrangian curves on spectral curves of monopoles

Abstract

We study Lagrangian points on smooth holomorphic curves in T${\mathbb P}^1$ equipped with a natural neutral K\"ahler structure, and prove that they must form real curves. By virtue of the identification of T${\mathbb P}^1$ with the space ${\mathbb L}({\mathbb E}^3)$ of oriented affine lines in Euclidean 3-space ${\mathbb E}^3$, these Lagrangian curves give rise to ruled surfaces in ${\mathbb E}^3$, which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in ${\mathbb E}^3$, called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in ${\mathbb E}^3$ where the number of oriented lines in the complex curve $\Sigma$ that pass through the point is less than the degree of $\Sigma$. We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves.