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Is it true that a minimiser of Finsler energy is automatically parameterised by arc length? As in the Riemannian case. Is there a reference for this fact?

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I don't know of a reference, but this is true and can be proved either directly or by reduction to the Riemannian case. Suppose $M$ is a Finsler manifold and $\gamma:[a,b]\to M$ is a curve. Then the pullback metric $ds=\|\gamma'(t)\|\,dt$ is a Riemannian metric on $[a,b]$. (In one dimension, any Finsler metric is a Riemannian metric). When we minimize over reparametrizations $\gamma\circ h$ with $h:[a,b]\to [a,b]$, we actually minimize the energy of $h$ as a map from $[a,b]$ with standard metric into $[a,b]$ with the pullback metric $ds$. It follows that the minimizer will have constant-speed parametrization (proportional to arc length).

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