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I'm trying to verify part of a claim from Freedman-He-Wang's paper on Mobius knot energy, in Prop. 8.3. My question is less about knot energies, and more about the preservation of length in a uniform limit. To be specific, suppose $\gamma_i: [0,1] \to \mathbb{R}^3$ is a sequence of rectifiable simple closed curves, each parametrized with respect to arc length and with total length $1$. Furthermore, suppose for each $\varepsilon > 0$, there is some $\delta > 0$ such that each $\gamma_i$ is $(1 + \varepsilon)$ bi-Lipschitz on subarcs of length $\leq \delta$. (This property is a consequence of Lemma 1.2, for knots with finite Mobius energy). Suppose $\gamma_i$ converges to a curve $\gamma$ uniformly. Must it be the case that $\gamma$ also has total length $1$?

Showing $L(\gamma) \leq 1$ is a straightforward application of Fatou's lemma. However, the uniform limit of curves could lose length. For instance, consider a sequence $\Gamma_i$ of loops of length $1$ each contained in concentric spheres of radius $\frac{1}{i}$. Then the uniform limit is a single point with zero length. This problem doesn't occur under our assumptions, but I'm having trouble showing the arc-length doesn't decrease in the limit.

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That is a beautiful paper. Regarding the question: Given $\epsilon>0$, consider the corresponding $\delta$, and take $n>1/\delta$. Then each $\gamma_i$ is $(1+\epsilon)$-bilipschitz on $[\frac{k-1}n,\frac{k}{n}]$. In particular, $$\forall i \ge 1, \quad \forall k=1,2\ldots, n , \quad \text{we have} \quad \Big|\gamma_i(\frac{k}{n})-\gamma_i(\frac{k-1}n)\Big| \ge \frac1{n(1+\epsilon)} \,.$$ Passing to the limit, $$ \forall k=1,2\ldots, n , \quad \text{we have} \quad \Big|\gamma (\frac{k}{n})-\gamma (\frac{k-1}n)\Big| \ge \frac1{n(1+\epsilon)} \,.$$ We infer that the length $L(\gamma)$ is at least $$\sum_{k=1}^n \Big|\gamma (\frac{k}{n})-\gamma (\frac{k-1}n)\Big| \ge \frac1{ 1+\epsilon } \,.$$ Since $\epsilon>0$ is arbitrary, we conclude that $L(\gamma)=1$.

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