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.
Best Answer
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...