I am from China. This is my first question in this website.
Given a family of embedded, smooth and closed curves $\{X_i(s)\}_{i=1}^\infty$ in a plane, let us denote by $L_i$ the perimeter and $\kappa_i(s)$ the relative curvature of $X_i$, respectively.
Suppose $\{X_i(s)\}_{i=1}^\infty$ satisfies the following:
(1) the area of the domain encloed by $X_i$ is $\pi$, $i=1, 2, \ldots$,
(2) $L_i$ tends to $+\infty$ as $i$ tends to infinity.
Can we conclude that $K_i\triangleq\max\{|\kappa_i(s)|\ |s\in[0, L_i]\}$ tends to $+\infty$ as $i$ tends to infinity? How to prove it if it is correct?
Thanks a lot!