  The doping and frequency evolutions of the incommensurate spin response and
the resonance mode are studied based on the scenario of the Fermi surface
topology. We use the slave-boson mean-field approach to the $t-t^{\prime}-J$
model and including the antiferromagnetic fluctuation correction in the
random-phase approximation. We find that the equality between the
incommensurability and the hole concentration is reproduced at low frequencies
in the underdoped regime. This equality observed in experiments was explained
{\it only} based on the stripe model before. We also obtain the downward
dispersion for the spin response and predict its doping dependence for further
experimental testing, as well as a proportionality between the low-energy
incommensurability and the resonance energy. Our results suggest a common
origin for the incommensuration and the resonance peak based on the Fermi
surface topology and the d-wave symmetry.
