Cambridge Journal of Mathematics

Volume 11 (2023)

Number 2

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

Pages: 299 – 439

DOI: https://dx.doi.org/10.4310/CJM.2023.v11.n2.a1

Authors

Nikolaos Kapouleas (Department of Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Peter McGrath (Department of Mathematics, North Carolina State University, Raleigh, N.C., U.S.A.)

Abstract

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 ${\lvert \breve{M} \rvert}$ of $\breve{M}$ in terms of the sizes of its catenoidal bridges and confirm that it is negative; ${\lvert \breve{M} \rvert} \lt 2 {\lvert \Sigma \rvert}$ 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.

Full Text (PDF format)

Received 10 September 2021

Published 6 June 2023