Thread-wire surfaces

Abstract

This thesis studies surfaces which minimize area, subject to a fixed boundary and to a free boundary with length constraint. Based on physical experiments, I make two conjectures. First, I conjecture that minimizers supported on generic wires have finitely many surface components. I approach this conjecture by proving that surface components of near-wire minimizers are Lipschitz graphs in wire Frenet coordinates, and appear near maxima of wire curvature. Second, I conjecture and prove that surface components of near-wire minimizers are C1 at corners where the thread touches the wire interior. Moreover, the limit of the surface normal field is the Frenet binormal of the wire at the corner point. This shows local wire geometry dominates global wire geometry in influencing the surface corner. Third, I show that these two conjectures are related: assuming additional regularity up to the corner, the finiteness conjecture follows.