  (NOTE: per referee comments, this article has been split; it is now
superseded by "Existence of thread-wire minimizers" and "Near-wire thread-wire
minimizers"; please see http://www.bkstephens.net.) Alt's thread problem asks
for least-area surfaces bounding a fixed "wire" curve and a movable "thread"
curve of length L. We conjecture that if the wire has finitely many maxima of
curvature, then its Alt minimizers have finitely many surface components. We
show that this conjecture reduces to controlling near-wire minimizers, and thus
begin a three paper series to understand them. In this paper we show they
arise, show that they are embedded, and show that they have a nice
parametrization in wire exponential coordinates. In doing so we prove tools of
independent interest: a weighted isoperimetric inequality, a nonconvex
enclosure theorem, and a classification of how Alt minimizers intersect planes.
The last item reduces to a question about harmonic functions in the spirit of
Rado's lemma.
