Feral curves and minimal sets

• 成果类型: Article
• 署名作者: Fish, Joel W.; Hofer, Helmut
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2023.197.2.2
• 发表日期: 2023
• 页码: 533-738
• 关键词: seifert conjecture; counterexamples; compactness; polyfolds

摘要

We prove that for each Hamiltonian function H is an element of C infinity(R4, R) defined on the standard symplectic (R4, omega 0), for which M := H-1(0) is a non-empty compact regular energy level, the Hamiltonian flow on M is not minimal. That is, we prove there exists a closed invariant subset of the Hamiltonian flow in M that is neither null nor all of M. This answers the four-dimensional case of a more than twenty year old question of Michel Herman, part of which can be regarded as a special case of the Gottschalk Conjecture. Our principal technique is the introduction and development of a new class of pseudoholomorphic curve in the symplectization R x M of framed Hamiltonian manifolds (M, lambda, omega). We call these feral curves because they are allowed to have infinite (so-called) Hofer energy, and hence may limit to invariant sets more general than the finite union of periodic orbits. Standard pseudoholomorphic curve analysis is inapplicable without energy bounds, and thus much of this paper is devoted to establishing properties of feral curves, such as area and curvature estimates, energy thresholds, compactness, asymptotic properties, etc.