NDLI: Lagrangian Curves on Spectral Curves of Monopoles

Content Provider	Springer Nature Link
Author	Guilfoyle, Brendan Khalid, Madeeha Ramón Marí, José J.
Copyright Year	2010
Abstract	We study Lagrangian points on smooth holomorphic curves in ${\rm T}{\mathbb P}^1$ equipped with a natural neutral Kähler structure, and prove that they must form real curves. By virtue of the identification of ${\rm T}{\mathbb P}^1$ 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 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.
Starting Page	255
Ending Page	273
Page Count	19
File Format	PDF
ISSN	13850172
Journal	Mathematical Physics, Analysis and Geometry
Volume Number	13
Issue Number	3
e-ISSN	15729656
Language	English
Publisher	Springer Netherlands
Publisher Date	2010-07-16
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Neutral Kaehler structure Monopoles Lagrangian curves Applications of Mathematics Group Theory and Generalizations Geometry Theoretical, Mathematical and Computational Physics Analysis
Content Type	Text
Resource Type	Article
Subject	Mathematical Physics Geometry and Topology

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