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)
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[1] Gromov, Three remarks on geodesic dynamics and fundamental group · Zbl 1002.53028
[2] Cheeger, Problems in Analysis pp 195– (1970)
[3] DOI: 10.1017/S0143385700002935 · Zbl 0572.58019 · doi:10.1017/S0143385700002935
[4] DOI: 10.2307/2373590 · Zbl 0254.58005 · doi:10.2307/2373590
[5] Schiffer, Functionals on Finite Riemannian Surfaces (1954)
[6] DOI: 10.1017/S0143385700001747 · Zbl 0525.58028 · doi:10.1017/S0143385700001747
[7] Hurder, Differentiability, rigidity and Godbillon-Vey classes for Anosov flows · Zbl 0725.58034 · doi:10.1007/BF02699130
[8] DOI: 10.1112/jlms/s2-24.2.351 · Zbl 0443.53035 · doi:10.1112/jlms/s2-24.2.351
[9] Ledrappier, Propriété de Poisson et courbure négative
[10] Klingenberg, Riemannische Geometrie in grossen (1968)
[11] Katok, Ergod. Th. & Dynam. Sys. 2 pp 339– (1982)
[12] Katok, AMS Trans. 116 pp 43– (1981)
[13] Katok, Publ. Math. IHES 51 pp 137– (1980) · Zbl 0445.58015 · doi:10.1007/BF02684777
[14] Margulis, Funct. Anal. Appl. 3 pp 69– (1969) · Zbl 0174.18704 · doi:10.1016/0022-1236(69)90051-2
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