  In Part I of this article we generalize the Linearized Doubling (LD)
approach, introduced in earlier work by NK, by proving a general theorem
stating that if $\Sigma$ is a closed minimal surface embedded in a Riemannian
three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a
suitable family of LD solutions on $\Sigma$, a minimal surface $\breve{M}$
resembling two copies of $\Sigma$ joined by many small catenoidal bridges can
be constructed by PDE gluing methods. (An LD solution $\varphi$ on $\Sigma$ is
a singular solution of the Jacobi equation with logarithmic singularities which
in the construction are replaced by catenoidal bridges.) We also determine the
first nontrivial term in the expansion for the area $|\breve{M}|$ of
$\breve{M}$ in terms of the sizes of its catenoidal bridges and confirm that it
is negative; $|\breve{M}| < 2 | \Sigma|$ follows.
  We demonstrate the applicability of the theorem by first constructing new
doublings of the Clifford torus. We then construct in Part II families of LD
solutions for general $(O(2)\times \mathbb{Z}_2)$-symmetric backgrounds
$(\Sigma, N,g)$. Combining with the theorem in Part I this implies the
construction of new minimal doublings for such backgrounds. (Constructions for
general backgrounds remain open.) This generalizes our earlier work for
$\Sigma=\mathbb{S}^2 \subset N=\mathbb{S}^3$ providing new constructions even
in that case.
  In Part III, applying the results of Parts I and II -- appropriately modified
for the catenoid and the critical catenoid -- we construct new self-shrinkers
of the mean curvature flow via doubling the spherical self-shrinker or the
Angenent torus, new complete embedded minimal surfaces of finite total
curvature in the Euclidean three-space via doubling the catenoid, and new free
boundary minimal surfaces in the unit ball via doubling the critical catenoid.
