A pseudo-gradient flow arising in contact form geometry. (English) Zbl 1336.53090
The author continues the study of the semi-flow first considered in [A. Bahri, Classical and quantic periodic motions of multiply polarized spin-particles, Pitman Research Notes in Mathematics Series. 378. Harlow: Longman (1998; Zbl 0891.58004)]. We give an outline of the setting in which the constructions take place, and refer to the previously mentioned work for motivations.
Consider a closed oriented contact three-manifold \(M\) admitting a pair of contact forms \(\alpha\) and \(\beta=\iota_\nu \alpha\), where \(\nu \in \ker \alpha\). The main example is the following. Take the unit cotangent bundle \(UT\Sigma\) of a surface \(\Sigma\), where \(\alpha\) is the canonical contact form induced by a choice of Riemannian metric on \(\Sigma\), and let \(\beta\) be given by \(i^*\alpha\) where \(i : T\Sigma \to T\Sigma\) denotes the bundle morphism rotating each fibre by \(\pi/2\) radians. In this case the vector field \(\nu\) is a suitable normalisation of the infinitesimal generator of the natural \(S^1\)-action on the fibres.
The semi flow considered by the author is defined on the Banach manifold \(C_\beta\) consisting of closed curves \(x \in H^1(S^1,M)\) in \(M\), smooth in the Sobolev sense, and satisfying \(\alpha(\dot{x})=c>0\) (the constant is allowed to depend on the curve) together with \(\beta(\dot{x})=0\), i.e., \(x\) is positively transverse with respect to \(\alpha\) and Legendrian with respect to \(\beta\). Note that the inclusion \(C_\beta \subset H^1(S^1,M)\) is a homotopy equivalence as shown by A. Maalaoui and V. Martino [Adv. Nonlinear Stud. 14, No. 2, 393–426 (2014; Zbl 1305.53079)] given the additional assumption that \(\nu\) “turns well”.
The critical points of the functional \(x \mapsto \int_x \alpha\) are the so-called periodic Reeb orbits of \(\alpha\). The constructed semi flow is a pseudo gradient of this functional restricted to \(C_\beta\). The interest of this flow comes from the fact that it limits to broken solutions consisting of pieces of Reeb orbits joined with pieces of Legendrian curves, both with respect to the contact form \(\alpha\).
Consider a closed oriented contact three-manifold \(M\) admitting a pair of contact forms \(\alpha\) and \(\beta=\iota_\nu \alpha\), where \(\nu \in \ker \alpha\). The main example is the following. Take the unit cotangent bundle \(UT\Sigma\) of a surface \(\Sigma\), where \(\alpha\) is the canonical contact form induced by a choice of Riemannian metric on \(\Sigma\), and let \(\beta\) be given by \(i^*\alpha\) where \(i : T\Sigma \to T\Sigma\) denotes the bundle morphism rotating each fibre by \(\pi/2\) radians. In this case the vector field \(\nu\) is a suitable normalisation of the infinitesimal generator of the natural \(S^1\)-action on the fibres.
The semi flow considered by the author is defined on the Banach manifold \(C_\beta\) consisting of closed curves \(x \in H^1(S^1,M)\) in \(M\), smooth in the Sobolev sense, and satisfying \(\alpha(\dot{x})=c>0\) (the constant is allowed to depend on the curve) together with \(\beta(\dot{x})=0\), i.e., \(x\) is positively transverse with respect to \(\alpha\) and Legendrian with respect to \(\beta\). Note that the inclusion \(C_\beta \subset H^1(S^1,M)\) is a homotopy equivalence as shown by A. Maalaoui and V. Martino [Adv. Nonlinear Stud. 14, No. 2, 393–426 (2014; Zbl 1305.53079)] given the additional assumption that \(\nu\) “turns well”.
The critical points of the functional \(x \mapsto \int_x \alpha\) are the so-called periodic Reeb orbits of \(\alpha\). The constructed semi flow is a pseudo gradient of this functional restricted to \(C_\beta\). The interest of this flow comes from the fact that it limits to broken solutions consisting of pieces of Reeb orbits joined with pieces of Legendrian curves, both with respect to the contact form \(\alpha\).
Reviewer: Georgios Dimitroglou Rizell (Cambridge)
MSC:
| 53D10 | Contact manifolds (general theory) |
| 55P35 | Loop spaces |
| 35A15 | Variational methods applied to PDEs |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
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