On the equality between the infimum obtained by solving various Plateau's problem

TL;DR

This paper compares different formulations of Plateau's problem, including  and singular homological boundary conditions, and demonstrates their equivalence in terms of infimum values.

Contribution

It establishes the equality of infimum solutions across various boundary condition formulations of Plateau's problem.

Findings

Different boundary condition approaches yield the same infimum value.

Comparison between homological boundary conditions and size-minimizing currents.

Confirmation of the equivalence of solutions in Plateau's problem.

Abstract

In this paper we will compare the Plateau's problem with \v{C}ech and singular homological boundary conditions, we also compare these with the size minimizing problem for integral currents with a given boundary. Finally we get the agreement on the infimum values for these Plateau's problem.