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Sharp systolic inequalities for Reeb flows on the three-sphere
Symplectic Geometry · Submitted 20 Apr 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our first main result is that $\rho_{\mathrm{sys}}\leq 1$ in a neighbourhood of the space of Zoll contact forms on $S^3$, with equality holding precisely at Zoll contact forms.
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The abstract, sentence by sentence
The systolic ratio of a contact form $\alpha$ on the three-sphere is the quantity \[ \rho_{\mathrm{sys}}(\alpha) = \frac{T_{\min}(\alpha)^2}{\mathrm{vol}(S^3,\alpha\wedge d\alpha)}, \] where $T_{\min}(\alpha)$ is the minimal period of closed Reeb orbits on $(S^3,\alpha)$. A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that $\rho_{\mathrm{sys}}\leq 1$ in a neighbourhood of the space of Zoll contact forms on $S^3$, with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a local middle-dimensional non-squeezing theorem, and a sharp systolic inequality for Finsler metrics on the two-sphere which are close to Zoll ones. Our second main result is that $\rho_{\mathrm{sys}}$ is unbounded from above on the space of tight contact forms on $S^3$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The systolic ratio of a contact form $\alpha$ on the three-sphere is the quantity \[ \rho_{\mathrm{sys}}(\alpha) = \frac{T_{\min}(\alpha)^2}{\mathrm{vol}(S^3,\alpha\wedge d\alpha)}, \] where $T_{\min}(\alpha)$ is the minimal period of closed Reeb orbits on $(S^3,\alpha)$. | background 1.00 | 0.09 | 0.03 | asserted 0.96 |
| s1 | A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. | background 1.00 | 0.04 | 0.02 | asserted 0.97 |
| s2 | Our first main result is that $\rho_{\mathrm{sys}}\leq 1$ in a neighbourhood of the space of Zoll contact forms on $S^3$, with equality holding precisely at Zoll contact forms. | result 1.00 | 0.95 | 0.69 | asserted 1.00 |
| s3 | This implies a particular case of a conjecture of Viterbo, a local middle-dimensional non-squeezing theorem, and a sharp systolic inequality for Finsler metrics on the two-sphere which are close to Zoll ones. | interpretation 1.00 | 0.79 | 0.11 | asserted 1.00 |
| s4 | Our second main result is that $\rho_{\mathrm{sys}}$ is unbounded from above on the space of tight contact forms on $S^3$. | result 1.00 | 0.94 | 0.33 | asserted 1.00 |
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The paper
Original abstract
The systolic ratio of a contact form $\alpha$ on the three-sphere is the quantity \[ \rho_{\mathrm{sys}}(\alpha) = \frac{T_{\min}(\alpha)^2}{\mathrm{vol}(S^3,\alpha\wedge d\alpha)}, \] where $T_{\min}(\alpha)$ is the minimal period of closed Reeb orbits on $(S^3,\alpha)$. A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that $\rho_{\mathrm{sys}}\leq 1$ in a neighbourhood of the space of Zoll contact forms on $S^3$, with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a local middle-dimensional non-squeezing theorem, and a sharp systolic inequality for Finsler metrics on the two-sphere which are close to Zoll ones. Our second main result is that $\rho_{\mathrm{sys}}$ is unbounded from above on the space of tight contact forms on $S^3$.
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How the literature uses it
2 arXiv papers cite this in 15 sentences
Papers citing it · 10 sentences without a confident label
Builds on it 1 paper
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Again, Proposition 3.1 in [ref] is stated for $r=1$, but the general case follows by rescaling.
Contact forms with large systolic ratio in dimension three · Construction of the plug · 0.99
Uses its method or tool 1 paper
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The last ingredient which we need is the following result from [ref] [Proposition 3.1], which allows us to lift an area preserving diffeomorphism of a disc to a Reeb flow on a solid torus.
Contact forms with large systolic ratio in dimension three · Construction of the plug · 0.89
Compares with it 2 papers
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The statement of [ref] is actually slightly more general, since it allows $λ_0$ to be replaced by a more general primitive of $dx∧dy$, and gives more properties of the contact form $β$.
Contact forms with large systolic ratio in dimension three · Construction of the plug · 0.92
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(This is similar to [ref] .
Mean action and the Calabi invariant · From a disk map to an open book · 0.93
What it cites · 6 sentences without a confident label
Builds on 1 paper
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The proof of the abve proposition is a simple modification of the proof of Theorem B.2 in [ref] , which deals with Zoll contact forms on $SO(3)$ instead of $S^3$.
A systolic inequality for geodesic flows on the two-sphere · Proof of Theorem [ref] · 0.86
Compares with 1 paper
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Unlike in the main result of [ref] , here we do not have an explicit description of the neighborhood $ℱ$ in terms of geometric quantities.
A systolic inequality for geodesic flows on the two-sphere · A systolic inequality for Finsler metrics on the two-sphere · 1.00
Cites as support 1 paper
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If the metric $F$ is Zoll, then $λ_F$ is a Zoll contact form on $S^*(S^2,F)≅SO(3)$, and the equality in ([ref]) holds by Theorem B.2 in [ref] and identity ([ref]).
A systolic inequality for geodesic flows on the two-sphere · Proof of Corollary [ref] · 0.96
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