Band renormalization effects (BRE) are comprehensively studied for a mixed state of dx2- y2-wave superconducting (d-SC) and antiferromagnetic (AF) orders, in addition to simple d-SC, AF, and normal (paramagnetic) states, by applying a variational Monte Carlo method to a two-dimensional Hubbard (t-t-U) model. In a weakly correlated regime (U=t ≲ 6), BRE are negligible on all the states studied. As previously shown, the effective band of d-SC is greatly renormalized but the modifications of physical quantities, including energy improvement, are negligible. In contrast, BRE on the AF state considerably affects various features of the system. Because the energy is markedly improved for t=t < 0, the AF state occupies almost the whole underdoped regime in phase diagrams. A doped metallic AF state undergoes a kind of Lifshitz transition at t = t L ∼ -0:05t as t/t varies, irrespective of the values of U/t and σ (doping rate). Pocket Fermi surfaces arise around π, 0) [(π/2,π/2)] for t > t L[t < t L], which corresponds to the electron-hole asymmetry observed in angle-resolved photoemission spectroscopy (ARPES) spectra. The coexistent state of the two orders is possible basically for t > t L, because the existence of Fermi surfaces near (π, 0) is a requisite for the electron scattering of q = (π,π). Actually, the coexistent state appears mainly for tL/t t' ≲ 0:2 in the mixed state. Nevertheless, the AF and coexisting states become unstable toward phase separation for -0:05 ≲ t/t ≲ 0:2 but become stable at other values of t=t owing to the energy reduction by the diagonal hopping of doped holes. We show that this instability does not directly correlate with the strength of d-SC.