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On spaces of Euclidean triangles and triangulated Euclidean surfaces

Ismail Saglam, Ken'Ichi Ohshika, Athanase Papadopoulos

TL;DR

This work constructs Thurston-style asymmetric metrics on spaces of Euclidean triangles and their triangulated surfaces, establishing two equivalent definitions: an edge-parameter ratio metric $\eta((A),(A'))$ and a Lipschitz-based description via best maps between boxes; both yield a Finsler structure with infinitesimal norm $F(A,v)=\max_i\{v_i/A_i\}$. It proves geodesic existence, characterizes geodesic paths (including bigeodesics), and provides multiple models (notably a 2D $(A_1,A_2)$-model) to understand the metric geometry. The authors extend the framework to spaces of singular Euclidean structures on surfaces with fixed triangulations, define analogous metrics and Finsler structures, and study topologies and completeness, giving Thurston-inspired completeness results and counterexamples. They further generalize to spaces of convex Euclidean polygons and discuss metrics obtained by averaging or taking the maximum over triangulations, analyzing their Finsler geometry and completeness behavior. Overall, the paper develops a robust theory of asymmetric, Finsler-type metrics for Euclidean triangulations and triangulated surfaces, with implications for Thurston-type deformation theories in Euclidean settings.

Abstract

In this paper, we introduce an asymmetric metric on the space of marked Euclidean triangles, and we prove several properties of this metric, including two equivalent definitions of this metric, one of them comparing ratios of functions of the edges, and the other one in terms of best Lipschitz maps. We give a description of the geodesics of this metric. We show that this metric is Finsler, and give a formula for its infinitesimal Finsler structure. We then generalise this study to the case of convex Euclidean polygons in the Euclidean plane and to surfaces equipped with singular Euclidean structures with an underlying fixed triangulation. After developing some elements of the theory of completeness and completion of asymmetric metrics which is adapted to our setting, we study the completeness of the metrics we introduce in this paper. These problems and the results obtained are motivated by Thurston's work developed in his paper Minimal stretch maps between hyperbolic surfaces. We provide an analogue of Thurston's theory in a Euclidean setting.

On spaces of Euclidean triangles and triangulated Euclidean surfaces

TL;DR

This work constructs Thurston-style asymmetric metrics on spaces of Euclidean triangles and their triangulated surfaces, establishing two equivalent definitions: an edge-parameter ratio metric and a Lipschitz-based description via best maps between boxes; both yield a Finsler structure with infinitesimal norm . It proves geodesic existence, characterizes geodesic paths (including bigeodesics), and provides multiple models (notably a 2D -model) to understand the metric geometry. The authors extend the framework to spaces of singular Euclidean structures on surfaces with fixed triangulations, define analogous metrics and Finsler structures, and study topologies and completeness, giving Thurston-inspired completeness results and counterexamples. They further generalize to spaces of convex Euclidean polygons and discuss metrics obtained by averaging or taking the maximum over triangulations, analyzing their Finsler geometry and completeness behavior. Overall, the paper develops a robust theory of asymmetric, Finsler-type metrics for Euclidean triangulations and triangulated surfaces, with implications for Thurston-type deformation theories in Euclidean settings.

Abstract

In this paper, we introduce an asymmetric metric on the space of marked Euclidean triangles, and we prove several properties of this metric, including two equivalent definitions of this metric, one of them comparing ratios of functions of the edges, and the other one in terms of best Lipschitz maps. We give a description of the geodesics of this metric. We show that this metric is Finsler, and give a formula for its infinitesimal Finsler structure. We then generalise this study to the case of convex Euclidean polygons in the Euclidean plane and to surfaces equipped with singular Euclidean structures with an underlying fixed triangulation. After developing some elements of the theory of completeness and completion of asymmetric metrics which is adapted to our setting, we study the completeness of the metrics we introduce in this paper. These problems and the results obtained are motivated by Thurston's work developed in his paper Minimal stretch maps between hyperbolic surfaces. We provide an analogue of Thurston's theory in a Euclidean setting.

Paper Structure

This paper contains 13 sections, 33 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. The space of Euclidean triangles
  3. Distance between triangles
  4. An alternative definition of the distance function
  5. Geodesics
  6. The Finsler structure on $\frak{T}_1$
  7. Another model for the space of triangles
  8. The case of a convex Euclidean structures on a surface with distinguished points
  9. Finsler structure on $\mathcal{E}(S,\mathcal{T})_1$
  10. The completion of an asymmetric metric space
  11. On the topology defined by the weak metric $\eta$
  12. Completeness
  13. Spaces of convex Euclidean polygons

Key Result

Proposition 2.1

The function $\eta$ is an asymmetric metric on $\frak{T}_1$.

Theorems & Definitions (81)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Definition 2.4: Geodesic
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • ...and 71 more