Eigenvalues and entropy of a Hitchin representation

By Rafael Potrie, Andr'es Sambarino. Inventiones Mathematicae, 2014

100 citations recorded by the source corpus.

We show that the critical exponent of a representation ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} in the Hitchin component of PSL(d,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{PSL}}}(d,\mathbb {R})$$\end{document} 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 ρ\X,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \backslash X,$$\end{document} where X is the symmetric space of PSL(d,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{PSL}}}(d,\mathbb {R})$$\end{document}. The proof relies in a construction useful to prove a regularity statement: if the Frenet equivariant curve of ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} is smooth, then ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} is Fuchsian.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a0c2ff5dc257c5843606321ab30b20751304ac5b

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Citations in the corpus

Positivity and higher Teichm\"uller theory (2018, unknown): “→ G/B which sends positive triples in RP1 to positive triples in G/B, where the notion of positivity of triples in G/B is given by Lusztig’s total positivity (see Section 3.1). We refer the reader to [39, 40, 41, 27, 29, 9, 45] for more geometric and dynamical properties of Hitchin representations. ”
Conformality for a robust class of non-conformal attractors (2019, methodology): “r more generally compact hyperbolic orbifolds). For these we show that hyperconvexity is also a closed condition (Proposition 9.3), and discuss a new proof and generalization of a result of Potrie-S. [43]. The two appendices of the paper rely on the Thermodynamical formalism: in Appendix A we extend”
An invitation to higher Teichm\"uller theory (2018, background): “Tholozan for n = 3 [109], and Potrie and Sambarino in general, establish an entropy rigidity theorem: A representation saturates the upper bound for the entropy if and only if it is a principal Fuchsian representation [100, 96].”
The geometry of maximal representations of surface groups into SO0(2,n) (2017, unknown): “or” of the length spectrum. Deﬁnition 1.5. Let ρ be a representation of Γ into SL(n,R). The entropy of ρ is the number h(ρ) = limsup R→+∞ 1 R log♯{γ ∈ [Γ] | Lρ(γ) ≤ R} . Theorem 1.6 (Potrie–Sambarino [PS17]). If ρ : Γ → SL(n,R) is a Hitchin representation, then h(ρ) ≤ 2 n −1 , with equality if and o”