  We generalize to non-flat geometries the formalism of Simon et al. (2005) to
reconstruct the dark energy potential. This formalism makes use of quantities
similar to the Horizon-flow parameters in inflation, can, in principle, be made
non-parametric and is general enough to be applied outside the simple, single
scalar field quintessence. Since presently available and forthcoming data do
not allow a non-parametric and exact reconstruction of the potential, we
consider a general parametric description in term of Chebyshev polynomials. We
then consider present and future measurements of H(z), Baryon Acoustic
Oscillations surveys and Supernovae type 1A surveys, and investigate their
constraints on the dark energy potential. We find that, relaxing the flatness
assumption increases the errors on the reconstructed dark energy evolution but
does not open up significant degeneracies, provided that a modest prior on
geometry is imposed. Direct measurements of H(z), such as those provided by BAO
surveys, are crucially important to constrain the evolution of the dark energy
potential and the dark energy equation of state, especially for non-trivial
deviations from the standard LambdaCDM model.

Astrophysics