[Paper Review] Singular Contact Geometry and Beltrami Fields in Cholesteric Liquid Crystals

Joseph Pollard, Gareth P. Alexander|arXiv (Cornell University)|Nov 22, 2019

Liquid Crystal Research Advancements36 references4 citations

TL;DR

This paper introduces singular contact structures—generalizations of contact geometry that allow isolated singularities—to model topological defects in cholesteric liquid crystals. By defining singularities in Beltrami fields and introducing a singular Lutz twist, the authors show all singular contact structures are homotopic within the singular class, prove non-removable singularities imply overtwistedness, and establish a singular version of the Weinstein conjecture, which holds when orbits connect singularities or form periodic orbits.

ABSTRACT

The description of point defects in chiral liquid crystals via topological methods requires the introduction of singular contact structures, a generalisation of regular contact structures where the plane field may have singularities at isolated points. We characterise the class of singularities that may arise in such structures, as well as the subclass of singularities that can occur in a Beltrami field. We discuss questions of global existence, and prove that all singular contact structures with nonremovable singularities are overtwisted. To connect the theory to experiment we also discuss normal and tangential boundary conditions for singular contact structures, and show we can realise all desired boundary conditions except for normal anchoring on a sphere, where a theorem of Eliashberg and Thurston provides an obstruction to having a singular contact structure in the interior. By introducing a singular version of the Lutz twist we show that all contact structures are homotopic within the larger class of singular contact structures. We give applications of our results to the description of topological defects in chiral liquid crystals.

Motivation & Objective

• To develop a mathematical framework for topological defects in chiral liquid crystals using singular contact structures.
• To characterize the types of singularities that can arise in Beltrami fields and singular contact structures.
• To address global existence and homotopy classification of singular contact structures, particularly in relation to overtwistedness.
• To connect the theory to physical boundary conditions in cholesteric systems, especially on spherical geometries.
• To formulate and investigate a singular version of the Weinstein conjecture, accounting for singularities in Reeb-like dynamics.

Proposed method

• Define singular contact structures as 2-plane fields on 3-manifolds with isolated point singularities, generalizing regular contact structures.
• Use the Beltrami field condition (curl v = λv) to link vector fields to contact forms via the 1-form dual to the vector field.
• Introduce the singular Lutz twist—a generalization of the classical Lutz twist—enabling homotopy within the space of singular contact structures.
• Apply techniques from confoliation theory (Eliashberg & Thurston) to analyze overtwistedness and topological obstructions.
• Construct counterexamples to the classical Weinstein conjecture using plug-like modifications near singularities, leading to the singular Weinstein conjecture.
• Analyze specific classes of fields (ABC fields, perturbations of harmonic 1-forms) to verify the singular conjecture in concrete cases.

Experimental results

Research questions

• RQ1 What types of singularities can occur in singular contact structures arising from Beltrami fields in cholesteric liquid crystals?
• RQ2 Can all singular contact structures be homotoped to one another within the class of singular contact structures?
• RQ3 Under what conditions do singular contact structures admit periodic orbits or orbits connecting singularities?
• RQ4 What topological obstructions arise for singular contact structures on closed manifolds, particularly spheres?
• RQ5 Does a singular version of the Weinstein conjecture hold, where Reeb-like fields must have either periodic orbits or orbits approaching singularities?

Key findings

• All singular contact structures with nonremovable singularities are overtwisted, generalizing a result from regular contact topology.
• The singular Lutz twist allows homotopy between any two contact structures within the larger class of singular contact structures, preserving homotopy type in the singular setting.
• A singular version of the Weinstein conjecture holds: every Reeb-like field of a singular contact form has either a closed periodic orbit or an orbit that limits on the singular set.
• For ABC fields, the singular Weinstein conjecture holds due to continuous deformation paths connecting singular to nonsingular fields via index-0 singularity unfoldings.
• Perturbations of closed 1-forms or intrinsically harmonic 1-forms yield singular contact forms whose Reeb-like fields have connecting orbits between singularities, satisfying the singular Weinstein conjecture.
• Normal anchoring on a sphere is obstructed for singular contact structures due to a theorem of Eliashberg and Thurston, indicating a fundamental topological constraint in spherical geometries.