We prove that for each Hamiltonian function $H\in \mathcal {C}^\infty (\mathbb {R}^4, \mathbb {R})$ defined on the standard symplectic $(\mathbb {R}^4, \omega _0)$, 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 $\emptyset $ 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. \par Our principal technique is the introduction and development of a new class of pseudoholomorphic curve in the “symplectization” $\mathbb {R} \times M$ 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.

我们证明对于定义在标准辛 $(\mathbb {R}^4, \omega _0)$，其中 $M:=H^{-1}(0)$ 是非空紧致正则能级，$M$ 上的哈密顿流不是最小的。也就是说，我们证明在 $M$ 中存在哈密顿流的封闭不变子集，它既不是 $\emptyset$ 也不是 $M$ 的全部。这回答了Michel Herman 二十多年前的一个问题的四维情况，其中一部分可以看作是Gottschalk 猜想的一个特例。\par 我们的主要技术是在框架哈密顿流形 $(M, \lambda , \omega )$ 的“辛化”$\mathbb {R} \times M$ 中引入和开发一类新的伪全纯曲线。我们称这些为野性曲线，因为它们被允许具有无限（所谓的）Hofer 能量，因此可能限制为比周期轨道的有限联合更一般的不变集。标准伪全纯曲线分析在没有能量界限的情况下是不适用的，因此本文的大部分内容都致力于建立自然曲线的性质，例如面积和曲率估计、能量阈值、紧凑性、渐近性质等。