Limits of Convex Projective Surfaces and Finsler Metrics

TL;DR

This paper demonstrates that growth rates of trace functions in certain convex projective surfaces can be described by a specific Finsler metric, confirming a conjecture and providing new insights into the geometry of the moduli space.

Contribution

It introduces a new Finsler metric with triangular unit balls to describe trace growth, confirming Loftin's conjecture with simpler proofs and broader applications.

Findings

Trace functions are approximated by lengths in a new Finsler metric.

The Finsler metric converges from Danciger and Stecker's asymmetric metric.

Results provide insights into compactifications of convex projective surface moduli spaces.

Abstract

We show that for certain sequences escaping to infinity in the $\operatorname{SL}_3\mathbb{R}$ Hitchin component, growth rates of trace functions are described by natural Finsler metrics. More specifically, as the Labourie-Loftin cubic differential gets big, logarithms of trace functions are approximated by lengths in a Finsler metric which has triangular unit balls and is defined directly in terms of the cubic differential. This is equivalent to a conjecture of Loftin from 2006 which has recently been proven by Loftin, Tamburelli, and Wolf, though phrasing the result in terms of Finsler metrics is new and leads to stronger results with simpler proofs. From our perspective, the result is a corollary of a more local theorem which may have other applications. The key ingredient of the proof is another asymmetric Finsler metric, defined on any convex projective surface, recently defined by Danciger and Stecker, in which lengths of loops are logarithms of eigenvalues. We imitate work of Nie to show that, as the cubic differential gets big, Danciger and Stecker's metric converges to our Finsler metric with triangular unit balls. While Loftin, Tamburelli, and Wolf address cubic differential rays, our methods also address sequences of representations which are asymptotic to cubic differential rays, giving us more insight into natural compactifications of the moduli space of convex projective surfaces.