Thread-wire surfaces: Near-wire minimizers and topological finiteness (superseded)

TL;DR

This paper investigates near-wire minimizers in Alt's thread problem, establishing their existence, embeddedness, and parametrization, and introduces new mathematical tools to analyze their properties, contributing to understanding minimal surfaces with boundary constraints.

Contribution

It demonstrates the existence and properties of near-wire minimizers, providing new tools like a weighted isoperimetric inequality and a classification of surface-plane intersections.

Findings

Near-wire minimizers are shown to exist and be embedded.

They admit a parametrization in wire exponential coordinates.

New mathematical tools are developed for analyzing minimal surfaces.

Abstract

(NOTE: per referee comments, this article has been split; it is now superseded by "Existence of thread-wire minimizers" and "Near-wire thread-wire minimizers"; please see http://www.bkstephens.net.)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.