Ginzburg-Landau equations on Riemann surfaces of higher genus
D. Chouchkov, N.M. Ercolani, S. Rayan, I.M. Sigal
TL;DR
This paper investigates the Ginzburg-Landau equations on higher genus Riemann surfaces, explicitly constructing solutions near constant curvature states, classifying related holomorphic structures, and analyzing energy properties.
Contribution
It provides explicit local moduli space constructions, a classification of holomorphic structures, and energy analysis for solutions on higher genus Riemann surfaces.
Findings
Explicit construction of local solutions near constant curvature states
Classification of holomorphic structures via degree, Abel-Jacobi map, and symmetric products
Identification of energy bounds relative to constant curvature solutions
Abstract
We study the Ginzburg-Landau equations on Riemann surfaces of arbitrary genus. In particular: - we construct explicitly the (local moduli space of gauge-equivalent) solutions in a neighbourhood of the constant curvature ones; - classify holomorphic structures on line bundles arising as solutions to the equations in terms of the degree, the Abel-Jacobi map, and symmetric products of the surface; - determine the form of the energy and identify when it is below the energy of the constant curvature (normal) solutions.
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Ginzburg–Landau equations on Riemann surfaces of higher genus
D. Chouchkov111Dept. of Math., U. of Toronto, Toronto, ON, M5S 2E4, Canada, N. M. Ercolani222Dept. of Math., U. of Arizona, Tucson, AZ, 85721-0089, USA, S. Rayan333Dept. of Math. & Stats., U. of Saskatchewan, Saskatoon, SK, S7N 5E6, Canada, I. M. Sigal444Dept. of Math., U. of Toronto, Toronto, ON, M5S 2E4, Canada
(April 23, 2019)