Transverse measures and best Lipschitz and least gradient maps. (English) Zbl 07902275
In this remarkable paper, the authors, motivated by work of Thurston on best Lipschitz maps between hyperbolic surfaces developed in [W. P. Thurston, “Minimal stretch maps between hyperbolic surfaces”, Preprint, arXiv:math/9801039], study infinity-harmonic maps from a hyperbolic manifold to the circle. They show that the best Lipschitz constant for such a map is supported on a geodesic lamination, as in Thurston’s setting. They also prove that in the case where the manifold is a surface, the dual problem leads to a function of least gradient which defines a transverse measure on the obtained lamination. They also discuss the construction of least gradient functions from transverse measures via primitives to Ruelle-Sullivan currents. The authors’ main contribution is that they introduce new ideas from geometric analysis into Thurston’s setting, shedding new light on an important part of the theory the latter developed.
Reviewer: Athanase Papadopoulos (Strasbourg)
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 30F60 | Teichmüller theory for Riemann surfaces |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 58E20 | Harmonic maps, etc. |
| 53C43 | Differential geometric aspects of harmonic maps |
Keywords:
\(p\)-harmoic maps; infinity-harmonic maps: best Lipschitz maps; laminations; transverse measures; Ruelle-Sullivan currentsReferences:
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