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Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g$.

At what conditions on $[\gamma_1] , \cdots, [\gamma_n]$ is any (isotopy class of) metric completely determined by the lengths of the geodesic representatives of respective classes $[\gamma_1] , \cdots, [\gamma_n]$ ?

I am not necessarily looking for a complete answer and partial answers/comments are very welcome.

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This paper is probably relevant for your question: MR0528966 Wolpert, Scott The length spectra as moduli for compact Riemann surfaces. Ann. of Math. (2) 109 (1979), no. 2, 323–351.

And here is a more recent paper on the subject:

MR3770180 Parlier, Hugo Interrogating surface length spectra and quantifying isospectrality. (English summary) Math. Ann. 370 (2018), no. 3-4, 1759–1787.

These papers suggest that finitely many geodesic lengths never determine the surface.

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  • $\begingroup$ Thanks for the references. Maybe I didn't insist enough on the fact that I want to work with isotopy classes of metrics, in which case we know that there exists at least one finite collection of homotopy classes whose lengths completely determine the metric (such a set can be made of cardinality 9g - 9 if I remember correctly). $\endgroup$ – Selim G Oct 21 at 18:13
  • $\begingroup$ The curves you refer to in your comment depend on the metric. In the original question you seem to fix the homotopy classes, and then consider all metrics. $\endgroup$ – Alexandre Eremenko Oct 22 at 10:34
  • $\begingroup$ The curves I work with in my comment do not depend on the metric (as in the question). $\endgroup$ – Selim G Oct 22 at 13:20

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