Quasiminimal sets and Plateau's problem with \v{C}ech homological conditions on $C^2$ submanifold

Abstract

Let $\Omega\subseteq \mathbb{R}^n$ be an $m$-dimensional closed submanifold of class $C^2$, $d$ be a positive integer between 1 and $m$. We will study the geometric and topological proprieties of quasiminimal sets in $\Omega$, and show that a minimizing sequence of $d$-sets converges to a minimal set in the sense of weak topology. Following from that, we can solve the Plateau's problem of dimension $d$ on $\Omega$ with \v{C}ech homological conditions.