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Four applications of conformal equivalence to geometry and dynamics. (English) Zbl 0668.58042
Conformal equivalence theorem from complex analysis says that every Riemannian metric on a compact surface with negative Euler characteristics can be obtained by multiplying a metric of constant negative curvature by a scalar function. This fact is used to produce information about the topological and metric entropies of the geodesic flow associated with a Riemannian metric, geodesic length spectrum, geodesic and harmonic measures of infinity and Cheeger asymptotic isoperimetric constant. The method is rather uniform and is based on a comparison of extremals for variational problems for conformally equivalent metrics.

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37A99 Ergodic theory
53A30 Conformal differential geometry (MSC2010)
Full Text: DOI
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