Peter McGrath

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 $\phi$ 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.

In earlier work of NK new closed embedded smooth minimal surfaces in the round three-sphere ${\mathbb{S}}^3(1)$ were constructed, each resembling two parallel copies of the equatorial two-sphere ${\mathbb{S}}_{eq}^2$ joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, or the equatorial circle and the poles. In this sequel we generalize those constructions so that the catenoidal bridges can concentrate along an arbitrary number of parallel circles, with the further option to include bridges at the poles. The current constructions follow the Linearized Doubling (LD) methodology developed before and the LD solutions constructed here can be modified readily for use to doubling constructions of rotationally symmetric minimal surfaces with asymmetric sides (work in progress). In particular they allow us to develop in this forthcoming work doubling constructions for the catenoid in Euclidean three-space, the critical catenoid in the unit ball, and the spherical shrinker of the mean curvature flow. Our constructions here allow for sequences of minimal surfaces where the catenoidal bridges tend to be "densely distributed", that is do not miss any open set of ${\mathbb{S}}_{eq}^2$ in the limit. This in particular leads to interesting observations which seem to suggest that it may be impossible to construct embedded minimal surfaces with isolated singularities by concentrating infinitely many catenoidal necks at a point.

In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to $O(n)$-invariant cones are rotationally symmetric.