31,449 research outputs found
Higher Order Schwarzians for Geodesic Flows, Moment Sequences, and the Radius of Adapted Complexifications
In the first part of the paper, comprising section 1 through 6, 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. With these functions and a
classical theorem of Loewner on analytic continuation we are able 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, equivalently for M compact, to the
existence of a Grauert tube of radius R in terms of infinite Hankel matrices
involving these Schwarzian functions. The basic characterization so obtained
can be expressed as a sequence of differential inequalities of increasing order
polynomial in the covariant derivatives of the Gauss curvature on M and in
{\pi}/R that should be regarded as the higher order versions of a curvature
inequality by L. Lempert and R. Sz\"oke. The second part of the paper, sections
7 through 11, includes a discussion of the rank of the infinite Hankel matrix
of the Schwarzians from part 1 and of new Schwarzians defined now for purely
imaginary radius, as well as some computations and examples. A characterization
of the existence of the adapted structure on T^RM in terms of moment sequences
with parameters R and v in TM is also noted
Inflaton field governed universe from NKK theory of gravity: stochastic approach
We study a nonperturbative single field (inflaton) governed cosmological
model from a 5D Noncompact Kaluza-Klein (NKK) theory of gravity. The inflaton
field fluctuations are estimated for different epochs of the evolution of the
universe. We conclude that the inflaton field has been sliding down its
(quadratic) potential hill along all the evolution of the universe and a mass
of the order of the Hubble parameter. In the model here developed the only free
parameter is the Hubble parameter, which could be reconstructed in future from
Super Nova Acceleration Probe (SNAP) data.Comment: accepted in European Physical Journal
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