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A new approach to quantum gravity is presented based on a nonlinearquantization scheme for canonical field theories with an implicitly definedHamiltonian. The constant mean curvature foliation is employed to eliminate themomentum constraints in canonical general relativity. It is, however, arguedthat the Hamiltonian constraint may be advantageously retained in the reducedclassical system to be quantized. This permits the Hamiltonian constraintequation to be consistently turned into an expectation value equation onquantization that describes the scale factor on each spatial hypersurfacecharacterized by a constant mean exterior curvature. This expectation valueequation augments the dynamical quantum evolution of the unconstrainedconformal three-geometry with a transverse traceless momentum tensor density.The resulting quantum theory is inherently nonlinear. Nonetheless, it isunitary and free from a nonlocal and implicit description of the Hamiltonianoperator. Finally, by imposing additional homogeneity symmetries, a broad classof Bianchi cosmological models are analyzed as nonlinear quantumminisuperspaces in the context of the proposed theory.Comment: 14 pages. Classical and Quantum Gravity (To appear

Nonlinear quantum gravity on the constant mean curvature foliation