Given a complete Riemannian manifold $\mathcal{M}\subset\mathbb{R}^d$ which is a Lipschitz neighbourhood retract of dimension $m+n$, of class $C^{3,\beta}$, without boundary and an oriented, closed submanifold $\Gamma \subset \mathcal M$ of dimension $m-1$, of class $C^{3,\alpha}$ with $\alpha<\beta$, which is a boundary in integral homology, we construct a complete metric space $\mathcal{B}$ of $C^{3,\alpha}$-perturbations of $\Gamma$ inside $\mathcal{M}$ with the following property. For the typical element $b\in\mathcal B$, in the sense of Baire categories, every $m$-dimensional integral current in $\mathcal{M}$ which solves the corresponding Plateau problem has an open dense set of boundary points with density $1/2$. We deduce that the typical element $b\in\mathcal{B}$ admits a unique solution to the Plateau problem. Moreover we prove that, in a complete metric space of integral currents without boundary in $\mathbb{R}^{m+n}$, metrized by the flat norm, the typical boundary admits a unique solution to the Plateau problem.

给定一个完备黎曼流形$\mathcal{M}\subset\mathbb{R}^d$，其中 是类的维度$m+n$的Lipschitz邻域收缩 $C^{3，\beta}$，无边界和有向闭子流形$\Gamma 维度为$m-1$的子集\数学M$，类为$C^{3，\Alpha}$ $\α<\β$，这是积分同调中的一个边界，我们构造了一个 $C^{3，\α}$的完备度量空间$\数学{B}$-$\Gamma$的扰动 具有以下属性的$\Mathcal{M}$内部。对于典型元素 在数学B$中，在Baire范畴的意义下，每一个$m$维 求相应平台的数学{M}中的积分电流 问题有一个密度为$1/2$的开的稠密边界点集。我们推论 数学{B}$中的典型元素$b\允许该问题的唯一解 高原问题。此外，我们还证明了在完备的积分度量空间中 以平坦范数度量的$\mathbb{R}^{m+n}$中的无边界电流， 典型边界为高原问题提供了唯一的解。