scholarly journals Admissible Hermitian metrics on families of line bundles over certain degenerating Riemann surfaces

2001 ◽  
Vol 197 (2) ◽  
pp. 441-489
Author(s):  
Wing-Keung To ◽  
Lin Weng
Keyword(s):  
Riemann Surfaces ◽  
Line Bundles ◽  
Hermitian Metrics

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

ASYMPTOTIC BEHAVIOR OF THE GINZBURG-LANDAU FUNCTIONAL ON COMPLEX LINE BUNDLES OVER COMPACT RIEMANN SURFACES

1996 ◽  
Vol 08 (03) ◽  
pp. 457-486
Author(s):  
GIANDOMENICO ORLANDI
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Chern Class ◽  
Riemann Surfaces ◽  
Line Bundles ◽  
Complex Line ◽  
Ginzburg Landau ◽  
Renormalized Energy ◽  
Compact Riemann Surfaces ◽  
Complex Line Bundles

Motivated by the works of F. Bethuel, H. Brezis, F. Hélein [5] and of F. Bethuel, T. Rivière [6], an asymptotic analysis is carried out for minimizers of the Ginzburg-Landau functional depending on a parameter ε, in the more general case of complex line bundles with prescribed Chern class over compact Riemann surfaces. Such a functional describes a 2-dimensional abelian Higgs model and is also related to phenomena in superconductivity. A suitable renormalized energy is defined which characterizes the singularities (degree one vortices) of the limiting configuration.

Topological Invariants and Differential Geometry

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Differential Geometry ◽  
Topological Space ◽  
Chern Class ◽  
Euler Number ◽  
Line Bundles ◽  
Topological Invariants ◽  
Fundamental Groups ◽  
Hermitian Metrics ◽  
Finite Set ◽  
First Chern Class

This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.

Extremal Hermitian metrics on Riemann surfaces with singularities

2000 ◽  
Vol 104 (2) ◽  
pp. 181-210 ◽  
Author(s):  
Xiaohua Zhu ◽  
Guofang Wang
Keyword(s):  
Riemann Surfaces ◽  
Hermitian Metrics
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