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Seminar

MLA - Weekly Seminar (Part 2): Local rigidity of manifolds with hyperbolic cusps November 14, 2019 (01:00 PM PST - 02:00 PM PST)
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Location: MSRI: Simons Auditorium
Speaker(s) Thibault Lefeuvre (Université de Paris XI)
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In 1985, Burns and Katok conjectured that the marked length spectrum of a closed negatively-curved manifold (i.e. the collection of lengths of closed geodesics marked by free homotopy) should determine the metric up to isometries. Otal and Croke independently proved the conjecture in 1990 for surfaces and despite some progress achieved in the 90s the problem did not really evolve until our recent proof of a local version of the conjecture obtained in collaboration with Guillarmou (the full conjecture is still an open question!). In 1995, Cao proved that Otal's method on surfaces can be generalized to deal with non-compact manifolds whose ends are real hyperbolic cusps, thus solving the Burns-Katok conjecture in this more general context. We will explain a proof of a local version of the Burns-Katok conjecture adapted to this setting. This involves (along with some geometric/dynamical systems arguments) a microlocal calculus introduced by Guedes Bonthonneau and inspired by Melrose's b-calculus in order to deal with the infinite ends of the manifold. Joint work with Yannick Guedes Bonthonneau.

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