The dynamics on three-dimensional strictly convex energy surfaces
type: Item
instance of: scholarly article
title: The dynamics on three-dimensional strictly convex energy surfaces
zbMATH Open document ID: 0944.37031
DOI: 10.2307/120994
author: Helmut Hofer
author: Krzysztof Wysocki
author: Eduard Zehnder
published in: Annals of Mathematics. Second Series
publication date: 2000-05-11T00:00:00Z
full work available at URL: 148 1.html
review text: This is an impressive piece of mathematics which to a large extent is based on the theory of pseudoholomorphic curves, developed by the same authors in previous work. Consider \({\mathbb R}^4\) with the standard symplectic form \(\omega = d\lambda_0\) where \(\lambda_0 = {1\over 2} (q_j dp_j - p_j dq_j)\), and let \(S\) be a compact energy surface \(H(x)=1\) of a regular Hamiltonian. Assuming \(H''(x) \geq a \text{ Id}\), the energy surface (which is assumed to contain the origin in its interior) is strictly convex. Such hypersurfaces \(S\) are known to carry at least 2 periodic orbits. One of the main theorems in this work states that there are either 2 or infinitely many such orbits. The real ``tour de force'' (from which this result is deduced) is the explicit construction of a global surface of section of \(S\), i.e. an embedded compact surface whose boundary components consist of periodic orbits and whose interior is suitably transversal to the flow. The desired surface of section requires singling out a special periodic solution, based on an index concept. Section 3 contains a detailed account of this theory and proves the following fundamental result. On a 3-dimensional manifold with contact form \(\lambda\) and associated Reeb vector field \(X_\lambda\), every contractible \(T\)-periodic solution \(x\) of \(X_\lambda\) gives rise to an integer \(\widetilde{\mu}(x,T)\); in the special case of a strictly convex hypersurface \(S\) of \({\mathbb R}^4\), equipped with the contact form \(\lambda_0|_S\), one has \(\widetilde{\mu}(x,T) \geq 3\). Since the property of being strictly convex is not symplectically invariant, the authors introduce the notion of a dynamically convex contact form \(\lambda\) on \(S^3\), which is one with the property \(\widetilde{\mu}\geq 3\) for every \(T\)-periodic solution of \(X_\lambda\). Assuming \(\lambda = f \lambda_0\) is dynamically convex (where \(f\) is a smooth positive function on \(S^3\)), the special periodic solution which…
reviewed by: W. Sarlet
Mathematics Subject Classification ID: 37K05
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37C27
Mathematics Subject Classification ID: 37J55
zbMATH DE Number: 1230554
zbMATH Keywords: strictly convex energy surfaces
zbMATH Keywords: global surface of section
zbMATH Keywords: contact form
zbMATH Keywords: periodic orbits
MaRDI profile type: Publication
OpenAlex ID: W2323637861
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description: scientific article; zbMATH DE number 1230554