Marked-length-spectral rigidity for flat metrics

Abstract

In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity.

Hyperbolic Structures on Surfaces and Geodesic Currents

“…The space of equivalence classes of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface S is denoted Flat(S). Associated to each metric ϕ ∈ Flat(S) is a geodesic current L ϕ called the Liouville current, which is formally a measure on the double boundary of the universal cover of S. We affirmatively answer an open question of Bankovic-Leininger showing that the support of L ϕ determines ϕ, up to affine deformation (see [BL18,Section 6]). That is, a flat metric is not only determined by its geodesic current but even by the support alone, which is quite different from the hyperbolic case where currents have full support.…”

“…Such currents exist for a wide range of metrics, generalizing the classical Liouville measure on geodesics in the hyperbolic plane given in terms of cross-ratios; see [Bon88, Ota90, Cro90, CFF92, HP97, DLR10, BL18, Con15]. The construction of the Liouville current for a flat metric and an investigation of its various properties was the focus of [DLR10] and [BL18]. We describe the key properties in §2.5, setting the stage for the more detailed analysis we will carry out in §3.…”

Marked length spectrum rigidity in nonpositive curvature with singularities

“…Outline of proof. Bankovic and Leininger introduce the notion of (L g , Ω)-chains ( [BL15,§4]). An (L g , Ω)-chain is a sequence (x i ) of points in ∂ ∞ (S) such that for all i,…”

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.…”