Inversion of Hopf's Umlaufsatz

Question

Hopf's Umlaufsatz (better known as?) says:

Let $\gamma$ be a simple closed differentiable positively oriented curve in the plane. Then for its curvature $\kappa$ it holds:

$$\int_{\gamma}\kappa\ \text{d}s = 2\pi$$

I wonder if (and cannot see why not) the inversion holds, too:

Let $\kappa$ be a continuous function $\kappa: [0,1] \rightarrow \mathbb{R}$ with $\kappa(0) = \kappa(1)$ and $\int_0^{1}\kappa\ \text{d}s = 2\pi$. Then there is a simple closed differentiable positively oriented curve $\gamma$ (of length 1) with curvature $\kappa$.

(If this holds, $\gamma$ would be unique upto congruency via Euclidean motions.)

Answer 1

There is another necessary condition on $\kappa$. The so-called Four vertex theorem:

Theorem: Suppose $\kappa$ is the curvature of a simple closed curve. Then, either $\kappa$ is constant or has at least 2 minimums and 2 maximums.

The ellipse shows that there is no reason to expect more than 2 minimums or maximums of curvature.

It turns out that together with your integral condition and this condition together are necessary and sufficient for $\kappa$ to correspond to a closed simple curve. A write up, due to DeTurck, Gluck, Pomerleano, and Vela-Vick, (who say the result is due to Dahlberg) can be found here. See also a more informal (but similar) write up by Vela-Vick here.