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Is it true that the length spectrum of a compact Riemannian manifold determines its volume?

This question was inspired by the MO question Length spectrum of spheres . BS's answer recalls a theorem of Duistermaat and Guillemin (Duistermaat, J. J.; Guillemin, V. W. (1975), "The spectrum of positive elliptic operators and periodic bicharacteristics", Inventiones Mathematicae 29 (1): 39–79) that states that for generic metrics the length spectrum and the spectrum of the Laplacian determine each other. Since Weyl's asymptotic formula implies that the spectrum of the Laplacian determines the volume, it may seem that the length spectrum determines the volume. Is this right?

Addendum. This question seems more useful interesting and useful if it is reformulated as follows: what does the set of lengths of periodic geodesics and/or its various analogues such as the length spectrum and the marked length spectrum say about the volume of a compact Riemannian or Finsler manifold?

Here are more concrete questions around this topic:

1. Let $M$ be a simply-connected, compact $n$-manifold. Does there exist a quantity $c(M) > 0$ such that for any Riemannian metric $g$ on $M$ the volume of $(M,g)$ is bounded below by $c(M)$ times the $n$-th power of the length of the shortest periodic geodesic of $(M,g)$?

This is a really hard open problem. It was solved by Croke for the $2$-sphere in Area and the length of the shortest closed geodesic J. Differential Geom. Volume 27, Number 1 (1988), 1-21. The current record for $c(S^2)$ is held by Rotman: The length of a shortest closed geodesic and the area of a -dimensional sphere Proc. Amer. Math. Soc. 134 (2006), 3041-3047. However, the conjectured optimal constant $1/2\sqrt{3}$ is still a challenge.

2. Are there known (explicit?) examples of compact Riemannian manifolds with the same set of lengths of periodic geodesics (the length set) and different volume?

BS gave a reference (see his answer) where Huber shows that two compact hyperbolic surfaces with the same length spectrum have the same laplace spectrum and hence the same volume. For negatively curved, compact surfaces J.-P. Otal showed that the marked length spectrum determines the surface up to isometries. This is not true for Finsler metrics (they are much more flexible than Riemannian metrics), however:

3. If two negatively-curved, compact Finsler surfaces have the same marked spectrum, do they also have the same area?

However, posing these type of question is easy. I'm more interested in collecting some success stories.

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Here's a relevant paper: ams.org/mathscinet-getitem?mr=2377017 I'm not sure what you mean by length spectrum - presumably with multiplicities (otherwise see this paper). On the other hand, what do you mean if there are infinitely many geodesics of the same length (such as the 2-sphere)? I guess you probably just want to restrict to generic metrics. See also: ams.org/mathscinet-getitem?mr=2097355 –  Ian Agol Mar 1 '12 at 17:14
    
Hi Agol, I mean with multiplicities. There is no problem with the sphere: for Zoll manifolds it is true that the length spectrum (the length of the prime closed geodesics) determines the volume. See my comment to the answer of BS. –  alvarezpaiva Mar 1 '12 at 17:19
    
So by "multiplicity", you mean if a manifold has infinitely many geodesics of a fixed length, you record the multiplicity as infinite? –  Ian Agol Mar 1 '12 at 23:38
    
There seems to be another notion of multiplicity in the literature: ams.org/mathscinet-getitem?mr=1306013 –  Ian Agol Mar 1 '12 at 23:46
    
This question seems way more complicated than I first thought. Indeed, one needs to give a precise definition of "length spectrum" that takes into account multiplicities, and there are tons of related questions than can be considered (and have been considered in the comments and answer). I'll look a bit more into this and edit the question to take your comments into account. To be continued ... –  alvarezpaiva Mar 2 '12 at 16:21

1 Answer 1

This seems like a difficult question, even for closed hyperbolic manifolds.

Indeed Marcos Salvai, in

"On the Laplace and complex length spectra of locally symmetric spaces of negative curvature." Math. Nachr. 239/240 (2002), plus erratum on his web page

proved that the complex length spectrum (with multiplicities) and the volume of a closed (oriented) hyperbolic manifold (real, complex, quaternionic or octonionic) determine the laplace spectrum (with multiplicities, even on forms) but cannot dispense with the volume. Hence even the complex length spectrum (length spectrum and holonomies) is not shown to determine volume. In the erratum, he seems to be confident that this can be repaired, but this is not published.

However, there is a recent preprint by Dubi Kelmer, which shows among other things, that the length spectrum determines the laplace spectrum (hence the volume) for compact hyperbolic manifolds (real, complex, quaternionic or octonionic). The methods seem strongly Lie group representation-theoretic, not generalizing to non locally symmetric (or homogeneous) manifolds.

Interestingly, he leaves open the question wether the laplace spectrum determines the multiplicities in the length spectrum (it is known that the length set is determined), whereas Salvai proves that the laplace-beltrami spectrum on forms determines the complex length spectrum, by following the proof by Gordon and Mao Math. Res. Lett. 1 (1994), no. 6, that it determines the length spectrum.

Complicated situation...

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Thanks BS. In the reference you give (the preprint by Kelmer) it is stated that Huber proved that for hyperbolic surfaces the length spectrum and the spectrum of the Laplacian determine each other (Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen, Math. Ann. 138 (1959) 1–26). Therefore, it is true that the length spectrum of a compact hyperbolic surface determines its volume. Another case where this happens is Zoll manifolds. This was known as the weak Blaschke conjecture and was settled by Yang and Reznikov. –  alvarezpaiva Mar 1 '12 at 17:15
    
I knew about hyperbolic surfaces, but somehow forgot to mention that this was known, and indeed since 1959, which I find quite remarkable. But the Zoll case is new to me. Thanks. –  BS. Mar 1 '12 at 18:50
    
For Euclidean manifolds, the Laplace spectrum does not determine the length spectrum. front.math.ucdavis.edu/0407.5422 –  Ian Agol Mar 1 '12 at 23:40

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