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arXiv · 2311.18510

Contact action functional, calculus of variation and canonical generating function of Legendrian submanifolds

Yong-Geun Oh·Seungook Yu
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Abstract

In the present paper, we formulate a contact analogue on the one-jet bundle $J^1B$ of Weinstein's observation which reads the classical action functional on the cotangent bundle is a generating function of any Hamiltonian isotope of the zero section. We do this by identifying the correct action functional which is defined on the space of Hamiltonian-translated (piecewise smooth) horizontal curves of the contact distribution, which we call the Carnot path space. Then we give a canonical construction of the Legendrian generating function which is the Legendrian counterpart of Laudenbach-Sikorav's canonical construction of the generating function of Hamiltonian isotope of the zero section on the cotangent bundle which utilizes a finite dimensional approximation of the action functional. Motivated by this construction, we develop a Floer theoretic construction of spectral invariants for the Legendrian submanifolds in the sequel [OY] which is the contact analog to the construction given in [Oh97, Oh99] for the Lagrangian submanifolds in the cotangent bundle.

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Yong-Geun Oh, Seungook Yu. 2023-11-30. Contact action functional, calculus of variation and canonical generating function of Legendrian submanifolds. https://arxiv.org/abs/2311.18510

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