Minimizing area subject to a length constraint
Benjamin Stephens
University of Toronto
January 29, 2007
Abstract: Alt's thread problem
concerns surfaces in $R^3$ which minimize area, subject to a fixed
boundary ("wire") and to a free boundary ("thread") with length
constraint. Alt minimizers typically consist of separate surface
components with corners where the thread pulls away from the
wire. I show that Alt minimizers lying near a generic wire are
$C^1$ near such corners. Moreover, they satisfy a very nice
property: the normals to the surface converge to the Frenet binormal of
the wire at the surface's corner point. This shows that at the
$C^1$ level, local wire geometry dominates global wire geometry in
influencing the surface corner. Moreover, nearby the Gauss map is
injective. Also, surface components of near-wire minimizers
appear only near maxima of wire curvature. This suggests an
approach to showing that Alt minimizers on generic wires have finitely
many surface components. For videos of related physical
experiments, see http://www.bkstephens.net.