  Self-consistent hydrodynamic one-loop quantum corrections to the
Gross-Pitaevskii equation due to the interaction of the condensate with
collective excitations are calculated. It is done by making use a formalism of
effective action and $\zeta$-function regularization for a contribution from
Bogoliubov particles in hydrodynamic approximation. It turns out to be possible
to reduce the problem to the investigation of a determinant of Laplace operator
on curved space where a metric is defined by density and velocity of the
condensate. Standard methods of quantum gravity let us get the leading
logarithmic contribution of the determinant and corresponding quantum
corrections. They describe an additional quantum pressure in the condensate,
local heating-cooling and evaporation-condensation effects of the
Bose-condensed fraction. The effects of these corrections are studied for the
correction from excited states in equilibrium situation. Response functions and
form factors are discussed in the same approximation.
