Harmonic maps for Hitchin representations

• Published in: Geometric and functional analysis Vol. 29; no. 2; pp. 539 - 560
• Main Author: Li, Qiongling
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2019; Springer Nature B.V
• Subjects: Analysis; Flux density; Mathematics; Mathematics and Statistics; Representations; Submerging
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-019-00491-7

Let ( S , g 0 ) be a hyperbolic surface, ρ be a Hitchin representation for P S L ( n , R ) , and f be the unique ρ -equivariant harmonic map from ( S ~ , g ~ 0 ) to the corresponding symmetric space. We show its energy density satisfies e ( f ) ≥ 1 and equality holds at one point only if e ( f ) ≡ 1 and ρ is the base n -Fuchsian representation of ( S , g 0 ). In particular, we show given a Hitchin representation ρ for P S L ( n , R ) , every ρ -equivariant minimal immersion f from the hyperbolic plane H 2 into the corresponding symmetric space X is distance-increasing, i.e. f ∗ g X ≥ g H 2 . Equality holds at one point only if it holds everywhere and ρ is an n -Fuchsian representation.