Abstract:
Let
$$ D:= \{ \gamma \in C^{3} ({\Bbb R} , {\Bbb R}^{3} ) |
\ \gamma (s)= \gamma(s+1), \ |\dot{\gamma}| \equiv 1
$$
$$
\gamma ([0,1]) \ is \ simple \ closed \ curve \}.
$$
In this paper we show that there is
$ \gamma \in D$ which minimizes the functional
$$
E_{\gamma_{0}} ( \gamma ) :=
\int_{0}^{1} | \ddot{ \gamma} (s) - \ddot{ \gamma_{0}} (s)| ^{2}ds
$$
$$
+ a \ area ( area \ minimizing \ surface \ with \ boundary \ \gamma ( [0,1]) )
$$
where $ \gamma_{0} \in D$ if $ a \in (0,\infty)$ is chosen suitably.
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