# Eigenvalues and entropy of a Hitchin representation

Eigenvalues and entropy of a Hitchin representation We show that the critical exponent of a representation $$\rho$$ ρ in the Hitchin component of $${{\mathrm{PSL}}}(d,\mathbb {R})$$ PSL ( d , R ) is bounded above, the least upper bound being attained only in the Fuchsian locus. This provides a rigid inequality for the area of a minimal surface on $$\rho \backslash X,$$ ρ \ X , where X is the symmetric space of $${{\mathrm{PSL}}}(d,\mathbb {R})$$ PSL ( d , R ) . The proof relies in a construction useful to prove a regularity statement: if the Frenet equivariant curve of $$\rho$$ ρ is smooth, then $$\rho$$ ρ is Fuchsian. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Inventiones mathematicae Springer Journals

# Eigenvalues and entropy of a Hitchin representation

, Volume 209 (3) – Feb 22, 2017
41 pages

/lp/springer_journal/eigenvalues-and-entropy-of-a-hitchin-representation-4ePHnwHtCJ
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0020-9910
eISSN
1432-1297
D.O.I.
10.1007/s00222-017-0721-9
Publisher site
See Article on Publisher Site

### Abstract

We show that the critical exponent of a representation $$\rho$$ ρ in the Hitchin component of $${{\mathrm{PSL}}}(d,\mathbb {R})$$ PSL ( d , R ) is bounded above, the least upper bound being attained only in the Fuchsian locus. This provides a rigid inequality for the area of a minimal surface on $$\rho \backslash X,$$ ρ \ X , where X is the symmetric space of $${{\mathrm{PSL}}}(d,\mathbb {R})$$ PSL ( d , R ) . The proof relies in a construction useful to prove a regularity statement: if the Frenet equivariant curve of $$\rho$$ ρ is smooth, then $$\rho$$ ρ is Fuchsian.

### Journal

Inventiones mathematicaeSpringer Journals

Published: Feb 22, 2017

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