Marked length spectrum rigidity in nonpositive curvature with singularities

Abstract

Combining several previously known arguments, we prove marked length spectrum rigidity for surfaces with nonpositively curved Riemannian metrics away from a finite set of cone-type singularities with cone angles > 2π. With an additional condition, we can weaken the requirement on one metric to 'no conjugate points.'

Hyperbolic cone metrics and billiards

“…There are other ways in which one could try to generalize the Current Support Theorem . As we mentioned in the introduction, not only are hyperbolic and Euclidean cone metrics determined by their Liouville currents, but so are any (variable) negative or non-positively curved cone metrics, by a result of Constantine [Con18]. Hence it is natural to ask to what extent they are determined by only their support: (Q6) Is there a version of the Current Support Theorem for negative or nonpositively curved cone metrics?…”

“…The proof follows Otal's approach [Ota90], associating to ϕ a Liouville (or Möbius) geodesic current, Lϕ which determines, and is determined by, the marked length spectrum, then proving that Lϕ determines ϕ; see also [Cro90,CFF92,Fra12]. The analogous result for unit area nonpositively curved Euclidean cone metrics-also called, flat metrics-was proved by the second author with Bankovic in [BL18] extending a special case with Duchin and Rafi [DLR10], and subsequently extended to all nonpositively curved Riemannian cone metrics by Constantine [Con18].…”

“…to be the set of all endpoints of basic geodesics. As mentioned in the introduction, the set G φ is exactly the support of the Liouville current Lϕ (see [BL18,Con18]). As a notational reminder, when some metric is involved, we will typically include the metric as a subscript to denote endpoints of geodesics while it will be included in parentheses to denote the actual geodesics.…”

Marked length rigidity for Fuchsian buildings

“…Of course, it is possible to formulate the marked length spectrum conjecture for other classes of geodesic spaces -for example, compact locally CAT(-1) spaces. Still in the realm of surfaces, Hersonsky and Paulin [HP97] extended the result to some singular metrics on surfaces, while Banković and Leininger [BL17] and Constantine [Con17] give extensions to the case of non-positively curved metrics. Moving away from the surface case, the conjecture was verified independently by Alperin and Bass [AB87] and by Culler and Morgan [CM87] in the special case of locally CAT(-1) spaces whose universal covers are metric trees.…”

Partial Marked Length Spectrum Rigidity Of Negatively Curved Surfaces

“…The theorem by Croke in [Cro90] is actually a result about surfaces of nonpositive curvature. Several works extended this result to more general types of metrics: to surfaces without conjugate points in [CFF92] and [Fat89], translation surfaces in [DLR10], flat surfaces with cone points in [BL18], and the combination of the previous cases in [Con18] . The marked length spectrum rigidity question has been studied in other contexts as well: in locally symmetric spaces of negative curvature in [Ham90], projective geometries in [Kim01] and [CD10], Fuchsian buildings in [CL19b], and 1-dimensional spaces in [CL19a].…”