Minimal surfaces on mirror-symmetric frames: A fluid dynamics analogy

Abstract

© The Author(s), 2020. Chaplygin's hodograph method of classical fluid mechanics is applied to explicitly solve the Plateau problem of finding minimal surfaces. The minimal surfaces are formed between two mirror-symmetric polygonal frames having a common axis of symmetry. Two classes of minimal surfaces are found: the class of regular surfaces continuously connecting the supporting frames forming a tube with complex shape; and the class of singular surfaces which have a partitioning film closing the tube in between. As an illustration of the general solution, minimal surfaces supported by triangular frames are fully described. The theory is experimentally validated using soap films. The general solution is compared with the known particular solutions obtained by the Weierstrass inverse method.