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Let $S$ be a soap film bounded by an unknotted wireframe cycle (in $R^3$). Why is it the case that as we deform the wireframe in $R^3$, $S$ deforms continuously?

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    $\begingroup$ and what about the phenomenon of continuous deformations of soap operas? $\endgroup$
    – Uri Bader
    Commented Oct 28, 2016 at 8:55

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You are asking whether a least area surface depends continuously on its boundary curve, or in mathematical terms: whether for every curve $\gamma$ there exists a least area surface $S$ with $\partial S=\gamma$ such that for any neighborhood of $S$ in the space of surfaces there exists a neighborhood of $\gamma$ in the space of curves such that for any curve $\gamma^\prime$ in that neighborhood one will find a least area surface $S^\prime$ with $\partial S^\prime=\gamma^\prime$ in the first neighborhood.

This is however not always the case, see

Beeson: Non-continuous dependence of surfaces of least area on the boundary curve. Pacific J. Math. 70 (1977), no. 1, 11–17.

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