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

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 $\mathcal{L}_\Sigma$ has trivial kernel, then given a suitable family of LD solutions on $\Sigma$, a minimal surface 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 linear equation $\mathcal{L}_\Sigma \varphi =0$ with logarithmic singularities; in the construction the singularities are replaced by catenoidal bridges.) As an example demonstrating the applicability of the theorem we construct new doublings of the Clifford torus. In Part II we construct families of LD solutions for general $(O(2)\times \Z_2)$-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 $\Sigma=\Sph^2 \subset N=\Sph^3$ providing new constructions even for that background. In Part III, applying the earlier result -- 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.