Computing Reeb dynamics on four-dimensional convex polytopes

Julian Chaidez; Michael Hutchings

doi:10.3934/jcd.2021016

Article Contents

2021, Volume 8, Issue 4: 403-445. Doi: 10.3934/jcd.2021016

This issue Previous Article Next Article Tracking the critical points of curves evolving under planar curvature flows

Computing Reeb dynamics on four-dimensional convex polytopes

Julian Chaidez and
Michael Hutchings

Department of Mathematics, University of California, Berkeley, CA 94720, USA

Received: September 2020

Revised: July 2021

Early access: August 2021

Published: October 2021

JC was partially supported by an NSF Graduate Research Fellowship. MH was partially supported by NSF grant DMS-1708899, a Simons Fellowship, and a Humboldt Research Award

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Abstract

We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio $ 1 $.

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Mathematics Subject Classification: Primary: 53D99, 37C25; Secondary: 53-08.

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Figure 1. We depict the tangent, normal and Reeb cones for two points $ p,q \in X $ in a polytope $ X \subset {\mathbb R}^2 $

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Figure 2. We depict sub-trajectories of the three types of orbits, in red. Each cube above represents a 3-face of a hypothetical 4-polytope

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Figure 3. An example of a flow graph with 4 nodes and 4 edges. The linear domains and flows are depicted above their corresponding nodes and edges

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