Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Ruben Jakob
Universität Tübingen
The „Thread Problem”: Its different analytic formulations, classical results and modern
methods
The classical, two-dimensional „Thread Problem” (Faden-Problem) P(Γ,L) is the problem to guarantee
the existence of some minimal surface X that partially spans some prescribed rectifiable Jordan-arc
Γ : [0,1] −→ R 3 , the „supporting wire”, and has partially free trace Σ, the „thread”, whose length is required
not to exceed some prescribed real number L >| Γ(0) − Γ(1) |> 0. Hence, P(Γ,L) is a free boundary
problem with a length constraint for the free trace Σ.
At first, the speaker plans to precisely explain this classical, parametric formulation as well as the modern
GMT-formulation of the n-dimensional „Thread Problem” P(Γ,L), which is to guarantee the existence
of some mass-minimizing integral n-current T within the class of all integral n-currents S on some R n+k
whose free boundaries Σ := ∂S − Γ have to meet the mass constraint M(Σ) ≤ L, for some prescribed real
value L and some fixed integral (n − 1)-current Γ on R n+k .
Secondly, the speaker wishes to present the most important existence and (partial) boundary regularity
results – concerning the „thread” Σ – about solutions of both types of the „Thread problem”, which have
been achieved so far, and to compare them to each other with respect to their physical meaning and
generality.
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