Cyclic surfaces and Hitchin components in rank 2

By F. Labourie. 2014

87 citations recorded by the source corpus.

We prove that given a Hitchin representation in a real split rank 2 group $\mathsf G_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin components by a Hermitian bundle over Teichm\"uller space. The proof goes through introducing holomorphic curves in a suitable bundle over the symmetric space of $\mathsf G_0$. Some partial extensions of the construction hold for cyclic bundles in higher rank.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/58f83f437228a429a80de0bcaa8e88224a2df28c

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In Geometry and Topology

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Citations in the corpus

The pressure metric for Anosov representations (2013, unknown): “D G ] 2 1”
An invitation to higher Teichm\"uller theory (2018, methodology): “For Hitchin components Labourie made a very precise conjecture, based on Hitchin’s parametrization of the Hitchin component.”
Eigenvalues and entropy of a Hitchin representation (2014, background): “let us remark that, as explained by Labourie [24, Section 1.4], the inequality in Theorem A implies a (rigid) inequality concerning the area of a minimal surface on ρ(π1Σ)\X.Theorem 1.4.1 of Labourie [24] hence holds for the Hitchin components of the groups G= PSL(d,R), PSp(2d,R), PSO(d,d+1) and the”
The geometry of maximal representations of surface groups into SO0(2,n) (2017, methodology): “Labourie’s conjecture was proven independently by Loftin [Lof01] and Labourie [Lab07] for G = SL(3,R), and then recently by Labourie [Lab17] for other split real Lie groups of rank 2 (namely, PSp(4,R) and G2).”
AN INVITATION TO HIGHER TEICHMÜLLER THEORY (2019, unknown): “François Labourie (2017b).”