Let $S^1=\mathbb R^1/\mathbb Z$. Consider a family $\varphi\_t$ of pieceswise smooth injective maps $\varphi\_t:S^1\to \mathbb C^1$ depending continuously on $t$. Then each curve $\varphi\_t(S^1)$ is a simple closed curve in $\mathbb C^1$, i.e. it bounds an open complex disk. Using Riemann mapping theorem we can identify such a disk with the unit disk $|z|\le 1$. In particular, for each $t$, by identifying the disk bounded by $\varphi\_t(S^1)$ with the disk $|z|\le 1$, we can define the cross-ratio of points $\varphi\_t(0)$, $\varphi\_t(\frac{1}{4})$, $\varphi\_t(\frac{1}{2})$, $\varphi\_t(\frac{3}{4})$ as points lying in $|z|\le 1$ . I am pretty sure that such a cross-ratio is a continuous function of $t$.


**Question.** Does this statement follow from some standard result? How to prove it?



Cross-ratios of $4$ boundary points on a continuous family of disks in $\mathbb C^1$
I realised that the answer to this question follows from a different question on Mathoverflow almost 10 years ago:


[Does Riemann map depend continuously on the domain?](https://mathoverflow.net/questions/51863/does-riemann-map-depend-continuously-on-the-domain)


These were good times... People were not voting to close questions just because they don't understand them...


