  Consideration in the present paper is the existence of global solutions for
the modified Camassa-Holm (mCH) equation with a nonzero background initial
value. The mCH equation is completely integrable and can be considered as a
model for the unidirectional propagation of shallow-water waves. By applying
the inverse scattering transform with an application of the Cauchy projection
operator, the existence of a unique global solution to the mCH equation in the
line with a nonzero background initial value is established in the weighted
Sobolev space $ H^{2, 1} (\mathbb{R})\cap H^{1, 2} (\mathbb{R})$ based on the
representation of a Riemann-Hilbert (RH) problem associated with the Cauchy
problem to the mCH equation. A crucial technique used is to derive the
boundedness of the solution in the Sobolev space $ W^{1,\infty}(\mathbb{R}),$
then reconstruct a new RH problem for the Cauchy projection operator of
reflection coefficients. The regularity of the global solution is achieved by
the refined estimate arguments on those solutions of the corresponding RH
problem.
