[Paper Review] An integer valued bi-invariant metric on the group of contactomorphisms of R^2n x S^1

Sheila Sandon|arXiv (Cornell University)|Oct 29, 2009

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper constructs an integer-valued bi-invariant metric on the group of compactly supported contactomorphisms of $\mathbb{R}^{2n} \times S^1$ that are isotopic to the identity, extending Viterbo's symplectic metric via generating functions. The metric is unbounded and compatible with the Bhupal partial order, demonstrating a new rigidity phenomenon in contact geometry with discrete energy values.

ABSTRACT

In his 1992 article on generating functions Viterbo constructed a bi-invariant metric on the group of compactly supported Hamiltonian symplectomorphisms of R^2n. Using the set-up of arXiv:0901.3112 we extend the Viterbo metric to the group of compactly supported contactomorphisms of R^2n x S^1 isotopic to the identity. We also prove that the contactomorphism group is unbounded with respect to this metric.

Motivation & Objective

• To extend Viterbo's symplectic bi-invariant metric to the contact setting on $\mathbb{R}^{2n} \times S^1$ using generating functions.
• To define a bi-invariant metric on $\mathrm{Cont}_0^c(\mathbb{R}^{2n} \times S^1)$ that takes only integer values.
• To prove that this metric is unbounded, showing the group is not bounded in this geometric structure.
• To establish compatibility between the metric and the Bhupal partial order, ensuring monotonicity of energy under positive isotopies.
• To clarify the role of integers in contact rigidity by constructing a discrete metric on a non-compact contact manifold.

Proposed method

• Adapts the generating function framework from [S09] to define invariants $c^+$ and $c^-$ for contactomorphisms of $\mathbb{R}^{2n} \times S^1$.
• Constructs the metric $d(\phi, \psi) = \lceil c^+(\phi \circ \psi^{-1}) \rceil + \lceil c^+(\psi \circ \phi^{-1}) \rceil$ using the integer-valued $c^+$ and $c^-$ invariants.
• Uses the fact that $c^+$ is non-negative and $c^-$ is non-positive for compactly supported contactomorphisms, which relies on the existence of points outside the support.
• Applies the energy-capacity inequality relating the metric to the contact capacity $c$ defined in [S09], ensuring geometric relevance.
• Proves unboundedness by constructing a sequence of contactomorphisms whose energy tends to arbitrarily large integers via increasing Hamiltonian growth.
• Demonstrates that the metric is not fine (since it takes discrete values) and not equivalent to the trivial norm, though it is not stably unbounded.

Experimental results

Research questions

• RQ1 Can Viterbo's symplectic bi-invariant metric be extended to the contactomorphism group of $\mathbb{R}^{2n} \times S^1$?
• RQ2 Does such an extension yield a metric that takes only integer values, reflecting the discrete nature of contact rigidity?
• RQ3 Is the resulting metric unbounded, indicating non-trivial geometric complexity in the contact group?
• RQ4 How does the metric interact with the Bhupal partial order, and does it preserve monotonicity of energy under positive isotopies?
• RQ5 Why does the construction fail in the 1-dimensional case ($n=0$), and what does this imply for the general theory?

Key findings

• The group $\mathrm{Cont}_0^c(\mathbb{R}^{2n} \times S^1)$ admits an integer-valued bi-invariant metric $d$ that is unbounded.
• The metric $d$ is constructed via generating functions and satisfies $d(\phi, \psi) = \lceil c^+(\phi \psi^{-1}) \rceil + \lceil c^+(\psi \phi^{-1}) \rceil$, with $c^+$ and $c^-$ defined using Legendrian lifts and generating functions.
• The metric is compatible with the Bhupal partial order $\leq_B$, making $(\mathrm{Cont}_0^c(\mathbb{R}^{2n} \times S^1), d)$ a partially ordered metric space.
• Energy does not decrease along isotopies generated by non-negative Hamiltonians, a consequence of the compatibility with $\leq_B$.
• The metric is unbounded because the energy of a sequence of contactomorphisms can be made arbitrarily large by increasing the growth of their generating Hamiltonians.
• The metric is not stably unbounded, as $\lim_{n \to \infty} \frac{E(\phi^n)}{n} = 0$ for all $\phi$, despite being unbounded.