Cartan's Magic Formula and Soap Film Structures

Details Cartan's Magic Formula and Soap Film Structures Cartan's Magic Formula and Soap Film Structures

2004-01-03

arxiv.org/abs/math/0401015v1

Mathematics

Classical Analysis and ODEs

AMS-LaTeX v1.2, 16 pages with 3 figures; uses birkart.cls

Scientific paper

A soap film is actually a thin solid fluid bounded by two surfaces of opposite orientation. It is natural to model the film using one polyhedron for each side. Two problems are to get the polyhedra for both sides to be in the same place without canceling each other out and to model triple junctions without introducing extra boundary components. We use chainlet geometry to create dipole cells and mass cells which accomplish these goals and model faithfully all observable soap films and bubbles. We introduce a new norm on chains of these cells and prove lower semicontinuity of area. A geometric version of Cartan's magic formula provides the necessary boundary coherence.

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