  In this paper, we construct time periodic doubly connected solutions for the
3D quasi-geostrophic model in the patch setting. More specifically, we prove
the existence of nontrivial $m$-fold doubly connected rotating patches
bifurcating from a generic doubly connected revolution shape domain with higher
symmetry $m\geq m_0$ and $m_0$ is large enough. The linearized matrix operator
at the equilibrium state is with variable and singular coefficients and its
spectral analysis is performed via the approach devised in [27] where a
suitable symmetrization has been introduced. New difficulties emerge due to the
interaction between the surfaces making the spectral problem richer and
involved.

Analysis of PDEs