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In Part I of this article we generalize the Linearized Doubling (LD)approach, introduced in earlier work by NK, by proving a general theoremstating that if $\Sigma$ is a closed minimal surface embedded in a Riemannianthree-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given asuitable family of LD solutions on $\Sigma$, a minimal surface $\breve{M}$resembling two copies of $\Sigma$ joined by many small catenoidal bridges canbe constructed by PDE gluing methods. (An LD solution $\varphi$ on $\Sigma$ isa singular solution of the Jacobi equation with logarithmic singularities whichin the construction are replaced by catenoidal bridges.) We also determine thefirst 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 itis negative; $|\breve{M}| < 2 | \Sigma|$ follows.  We demonstrate the applicability of the theorem by first constructing newdoublings of the Clifford torus. We then construct in Part II families of LDsolutions for general $(O(2)\times \mathbb{Z}_2)$-symmetric backgrounds$(\Sigma, N,g)$. Combining with the theorem in Part I this implies theconstruction of new minimal doublings for such backgrounds. (Constructions forgeneral backgrounds remain open.) This generalizes our earlier work for$\Sigma=\mathbb{S}^2 \subset N=\mathbb{S}^3$ providing new constructions evenin that case.  In Part III, applying the results of Parts I and II -- appropriately modifiedfor the catenoid and the critical catenoid -- we construct new self-shrinkersof the mean curvature flow via doubling the spherical self-shrinker or theAngenent torus, new complete embedded minimal surfaces of finite totalcurvature in the Euclidean three-space via doubling the catenoid, and new freeboundary minimal surfaces in the unit ball via doubling the critical catenoid.

Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers