Boundary Currents of Hitchin Components

2025/04/16 by Reid, Charles
Mathematics · Physics and Astronomy · #14M35 #51N15 #57K20 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2504.12294

openalex publication_date 2025/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The space of Hitchin representations of the fundamental group of a closed surface S into SL_nℝ embeds naturally in the space of projective oriented geodesic currents on S. We find that currents in the boundary have combinatorial restrictions on self-intersection which depend on n. We define a notion of dual space to an oriented geodesic current, and show that the dual space of a discrete boundary current of the SL_nℝ Hitchin component is a polyhedral complex of dimension at most n-1. For endpoints of cubic differential rays in the SL₃ℝ Hitchin component, the dual space is the universal cover of S, equipped with an asymmetric Finsler metric which records growth rates of trace functions.

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