Remarks on the oscillation energy of Legendrian isotopies
Dylan Cant
TL;DR
This paper constructs specific contact manifolds with Legendrian submanifolds that can be displaced with minimal oscillation energy, revealing degeneracy in a related pseudo-metric and exploring properties of Legendrian isotopies.
Contribution
It introduces new examples of Legendrian submanifolds with low oscillation energy displacement and analyzes the implications for the Shelukhin-Chekanov-Hofer pseudo-metric.
Findings
Existence of Legendrians displaced with arbitrarily small oscillation energy
Degeneracy of the Shelukhin-Chekanov-Hofer pseudo-metric on certain Legendrian classes
Insights into the oscillation energy of Legendrian isotopies
Abstract
We construct non-compact contact manifolds containing compact Legendrians which can be displaced from their Reeb flow with arbitrarily small oscillation energy. We use this to show the Shelukhin-Chekanov-Hofer pseudo-metric considered by Rosen and Zhang is degenerate on the isotopy class of the constructed Legendrians. Other aspects related to oscillation energy of Legendrian isotopies are explored.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Mathematical Dynamics and Fractals · Geometry and complex manifolds