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This paper is a part of a series devoted to the Euclidean-hyperboloidalfoliation method introduced by the authors for investigating the globalexistence problem associated with nonlinear systems of coupledwave-Klein-Gordon equations with small data. This method was developedespecially for investigating the initial value problem for the Einstein-massivefield system in wave gauge. Here, we study the (fourth-order) field equationsof f(R) modified gravity and investigate the global dynamical behavior of thegravitational field in the near-Minkowski regime. We establish the existence ofa globally hyperbolic Cauchy development approaching Minkowski spacetime (inspacelike, null, and timelike directions), when the initial data set issufficiently close to an asymptotically Euclidean and spacelike hypersurface inMinkowski spacetime. We cast the (fourth-order) f(R) field equations in theform of a second-order wave-Klein-Gordon system, which has an analogousstructure to the Einstein-massive field system but, in addition, involves a(possibly small) effective mass parameter. We establish the nonlinear stabilityof the Minkowski spacetime in the context of $f(R)$ gravity, when the integrandf(R) in the action functional can be taken to be arbitrarily close to theintegrand R of the standard Hilbert-Einstein action.

The Euclidean-hyperboloidal foliation method. Application to f(R) modified gravity