The minimal curvature of the universe in modified gravity and conformal anomaly resolution of the instabilities

We discuss the modified gravity which may produce the current cosmic acceleration of the universe and eliminate the need for dark energy. It is shown that such models where the action quickly grows with the decrease of the curvature define the FRW universe with the minimal curvature. Infinite time is required to reach the minimal curvature during the universe evolution. It is demonstrated that quantum effects of conformal fields may strongly suppress the instabilities discovered in modified gravity. We also briefly speculate on the modification of gravity combined with the presence of the cosmological constant dark energy.