Weak form of equidistribution theorem for harmonic measures of foliations by hyperbolic surfaces (Q2790205)

The author considers the existence of a harmonic measure on a smooth closed manifold which is foliated by a family of hyperbolic surfaces, i.e., a 2-dimensional smooth foliation equipped with a smooth leafwise metric of constant curvature \(-1\). By a counterexample in Section 2 of the paper, the author shows that the equidistribution theorem of \textit{C. Bonatti} and \textit{X. Gómez-Mont} [in: Essays on geometry and related topics. Mémoires dédiés à André Haefliger. Vol. 1. Genève: L'Enseignement Mathématique. 15--41 (2001; Zbl 1010.37025)] does not hold for foliations by hyperbolic surfaces (Theorem 2.1). The new and excellent result of the paper is introduced in Section 3. Specifically, the author introduces a weak form of the equidistribution theorem. More precisely, he shows the existence of a harmonic measure of folations by hyperbolic surfaces (Theorem 3.1). The author also introduces one further question in the last section.