Lagrangian Curves on Spectral Curves of Monopoles

Authors

Brendan Guilfoyle, Department of Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland.Follow
Madeeha Khalid, Department of Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland.Follow
Jose J. Ramon Marı, IT Tralee Research Institute, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland.Follow

Document Type

Article

Publication Details

Mathematical Physics, Analysis and Geometry

© 2010 The Authors.

Abstract

We study Lagrangian points on smooth holomorphic curves in TP 1 equipped with a natural neutral K¨ahler structure, and prove that they must form real curves. By virtue of the identification of TP 1 with the space L(E 3 ) of oriented affine lines in Euclidean 3-space E 3 , these Lagrangian curves give rise to ruled surfaces in E 3 , which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in E 3 , called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in E 3 where the number of oriented lines in the complex curve Σ that pass through the point is less than the degree of Σ. 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.

Recommended Citation

Guilfoyle, B., Khalid, M. & Ramón Marí, J.J. Lagrangian Curves on Spectral Curves of Monopoles. Math Phys Anal Geom 13, 255–273 (2010). https://doi.org/10.1007/s11040-010-9078-7