scholarly journals Energy of twisted harmonic maps of Riemann surfaces

2007 ◽  
pp. 45-61 ◽  
Author(s):  
William M. Goldman ◽  
Richard A. Wentworth
Keyword(s):  
Riemann Surfaces ◽  
Harmonic Maps

GEOMETRY OF THE MODULI SPACES OF HARMONIC MAPS INTO LIE GROUPS VIA GAUGE THEORY OVER RIEMANN SURFACES

2001 ◽  
Vol 12 (03) ◽  
pp. 339-371
Author(s):  
MARIKO MUKAI-HIDANO ◽  
YOSHIHIRO OHNITA
Keyword(s):  
Critical Point ◽  
Lie Groups ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Energy Function ◽  
Riemann Surfaces ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Point Subset ◽  
Presymplectic Structure

This paper aims to investigate the geometry of the moduli spaces of harmonic maps of compact Riemann surfaces into compact Lie groups or compact symmetric spaces. The approach here is to study the gauge theoretic equations for such harmonic maps and the moduli space of their solutions. We discuss the S1-action, the hyper-presymplectic structure, the energy function, the Hitchin map, the flag transforms on the moduli space, several kinds of subspaces in the moduli space, and the relationship among them, especially the structure of the critical point subset for the energy function on the moduli space. As results, we show that every uniton solution is a critical point of the energy function on the moduli space, and moreover we give a characterization of the fixed point subset fixed by S1-action in terms of a flag transform.

Harmonic maps from Riemann surfaces into complex Finsler manifolds

2015 ◽  
Vol 26 (06) ◽  
pp. 1541010
Author(s):  
Seiki Nishikawa
Keyword(s):  
Riemann Surfaces ◽  
Harmonic Maps ◽  
Positive Curvature ◽  
Harmonic Map ◽  
Riemann Sphere ◽  
Energy Functional ◽  
Finsler Metric ◽  
Absolute Minimum ◽  
Holomorphic Quadratic Differential ◽  
Smooth Map

Given a smooth map from a compact Riemann surface to a complex manifold equipped with a strongly pseudoconvex complex Finsler metric, we define the [Formula: see text]-energy of the map, whose absolute minimum is attained by a holomorphic map. A harmonic map is then defined to be a stationary map of the [Formula: see text]-energy functional. We prove that with each harmonic map is associated a holomorphic quadratic differential on the domain, which vanishes if the map is weakly conformal. Also, under the condition that the metric be weakly Kähler, we determine the second variation of the functional, and prove that any [Formula: see text]-energy minimizing harmonic map from the Riemann sphere to a weakly Kähler Finsler manifold of positive curvature is either holomorphic or anti-holomorphic.

scholarly journals One-harmonic maps on Riemann surfaces

1995 ◽  
Vol 3 (4) ◽  
pp. 645-681 ◽  
Author(s):  
Stefano Trapani ◽  
Giorgio Valli
Keyword(s):  
Riemann Surfaces ◽  
Harmonic Maps
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