  In this contribution, we summarize our recent studies of the phase structure
of the Grosse-Wulkenhaar model and its connection to renormalizability. Its
action contains a special term that couples the field to the curvature of the
noncommutative background space. We first analyze the numerically obtained
phase diagram of the model and its three phases: the ordered, the disordered,
and the noncommutative stripe phase. Afterward, we discuss the analytical
derivation of the effective action and the ordered-to-stripe transition line,
and how the obtained expression successfully explains the curvature-induced
shift of the triple point compared to the model without curvature. This shift
also causes the removal of the stripe phase and makes the model renormalizable.
