Geometry Workshop Obergurgl 2021

Over many decades, soap films have been studied in many fields. In physics, soap films are multi-layered fluids governed by the Navier-Stokes equations. The mathematics community, on the other hand, has investigated the geometry of their steady states as they have minimal areas among possible film surfaces. In this work, we bridge these two points of view. We propose a reformulation of soap film dynamics as a geometric flow. Our equation describes the dynamics of film surfaces while naturally leading them to the steady states in a locally area-minimized configuration. We also show another connection between geometry and dynamics; we define the equation purely from a geometric consideration, but it can also be derived from the Navier-Stokes equations. We finally present a numerical scheme for our geometric flow. This serves as a simulation method for soap films and a numerical solver for Plateau's problem for an arbitrary union of surfaces and enclosed volumes. Since our formulation naturally handles external forces like gravity, it provides a framework to investigate a more generalized form of Plateau's problem with the presence of external forces.