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Non-smooth geodesic flows and the earthquake flow on Teichmüller space. (English) Zbl 0692.57004

The author shows that earthquake semi-flows on any Teichmüller space \({\mathcal T}\) are not \({\mathcal C}^ 3\) smooth, where the flows are parametrized as in a paper of S. Kerckhoff [Ann. Math., II. Ser. 117, 235-265 (1983; Zbl 0528.57008)]. This is accomplished by deriving the differential equations governing the associated semi-flows on the tangent bundle T\({\mathcal T}\) to \({\mathcal T}\) in terms of quantities \(\Gamma^ k_{ij}\) which transform as Christoffel symbols but are defined on T\({\mathcal T}\) itself. An elementary argument (abstracted to the setting of certain flows with potentially low regularity) shows that \({\mathcal C}^ 2\) smoothness of the infinitesimal generator implies that the \(\Gamma^ k_{ij}\) depend only on the underlying point of \({\mathcal T}\). The assumption of \({\mathcal C}^ 2\) smoothness then allows the explicit computation of the \(\Gamma^ k_{ij}\) for the cases of the once-punctured torus and the torus-minus-a-disk; certain earthquake flows for these surfaces are found to violate convexity of hyperbolic lengths under earthquakes [see S. Kerckhoff, loc. cit.], and the proof for a general surface follows directly from the latter case. Certain standard material on measured geodesic laminations and earthquakes is briefly surveyed.
Reviewer: R.Penner

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
57R30 Foliations in differential topology; geometric theory
57R50 Differential topological aspects of diffeomorphisms
30F99 Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Citations:

Zbl 0528.57008
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