Covers, soap films and BV functions

TL;DR

This paper reviews the covering space method for solving Plateau's problem using BV functions and analyzes complex soap film examples, highlighting the method's applicability and limitations.

Contribution

It introduces a detailed analysis of soap film configurations using the covering space approach, expanding understanding of minimal surface solutions.

Findings

Covering space method effectively models certain soap films.

Some soap films, like retracting ones, are not modelable with Reifenberg.

Examples demonstrate the method's ability to handle complex boundary conditions.

Abstract

In this paper we first review the covering space method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with the covering space method: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, aa soap film that retracts onto its boundary, hence not modelable with the Reifenberg method, and various soap films spanning an octahedral frame.