  Homogeneous cosmological models with non-vanishing intrinsic curvature
require a special treatment when they are quantized with loop quantum
cosmological methods. Guidance from the full theory which is lost in this
context can be replaced by two criteria for an acceptable quantization,
admissibility of a continuum approximation and local stability. A quantization
of the corresponding Hamiltonian constraints is presented and shown to lead to
a locally stable, non-singular evolution compatible with almost classical
behavior at large volume. As an application, the Bianchi IX model and its
modified behavior close to its classical singularity is explored.
