HIGHER ORDER SCHWARZIANS FOR GEODESIC FLOWS, MOMENT SEQUENCES AND THE RADIUS OF ADAPTED COMPLEXIFICATIONS

In the first part of the paper, comprising Sections 1–5, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order Schwarzians of the linearized geodesic flow. We use these functions and a classical theorem of Loewner on analytic continuation to characterize the existence of the adapted complex structure induced by g on the set T^RM of vectors in TM of length up to R, that is, for M compact, the existence of a Grauert tube of radius R. The basic characterization obtained is expressed as a sequence of differential inequalities of increasing order polynomial in the derivatives of the Gauss curvature on M and in π R⁻¹ that should be regarded as the higher order versions of a curvature inequality by Lempert and Szőke. The second part of the paper, Sections 6–9, includes a discussion of the rank of the infinite Hankel matrix of the Schwarzians. In addition, a characterization of the existence of the adapted complex structure on T^RM in terms of moment sequences with parameters R and v in TM is noted.