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May 11, 2021: The Labourie Conjecture


The non-abelian Hodge correspondence builds a bridge between the character variety of surface group representations into a real Lie group G and the moduli space of G-Higgs bundles on a Riemann surface. The Labourie Conjecture, formulated as a question nearly fifteen years ago, addresses one of the most vexing aspects of this correspondence, namely its failure to transport the action of the mapping class group on character varieties; when true the conjecture provides a remedy for this deficiency on the Higgs bundle side. In recent years several special cases had been identified where the conjecture is true but a new result announced last month implies that the conjecture cannot be true in its most general form.


Note: All times are in (UTC-4)

Talk Titles/Abstracts

Brian Collier

Title: The minimal surface is unique (in rank 2)
Abstract: We will discuss all known cases where (an appropriate generalization) of the statement of Labourie's conjecture is true. In each known case, the rank of the symmetric space is 2 and the proof of uniqueness relies on a "rank 1" phenomenon of a related space.

Presentation slides available here.

Francois Labourie

Title: Uniformization for Hitchin representations

Abstract: After recalling the definition of Hitchin representations, and why the space of Hitchin representations carries many similarities with Teichmüller space, I will explain briefly Hitchin parametrisations and how minimal surfaces enterthe picture naturally, as well as some properties of the natural map from thespace of equivariant minimal surfaces into the Hitchin component.

I will then explain the main itching question that the conjecture was supposed to solve: find a complex interpretation, invariant under the mapping class group,of the Hitchin component, as well as describing the topology of the Hitchin moduli space.If time permits, I will explain a construction by Sambarino and myself, of a mapping classgroup invariant map from Hitchin component to Teichmüller space and how thisrelates to holomorphic differentials.

Presentation slides available here.

Vlad Markovic  

Title: Non-uniqueness of minimal surfaces in products of closed Riemann surfaces
Abstract: We show that there exists a Fuchsian representation into the product of three copies of PSL(2,R) which yields multiple minimal surfaces in the corresponding product of closed Riemann surfaces.

Presentation slides available here.


We will meet and mingle in the gathertown venue and will access the talks on
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Meeting ID: 862 1661 9087

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