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.
For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from classical Deligne–Lusztig theory by using the loop space functor. We study this construction in the special case that $G$ is an inner form of $\mathrm{GL}_n$ and the loop Deligne–Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its $\ell$-adic cohomology realizes many irreducible supercuspidal representations of $G$, notably almost all among those whose L‑parameter factors through an unramified elliptic maximal torus of $G$. This gives a purely local, purely geometric and—in a sense—quite explicit way to realize special cases of the local Langlands and Jacquet–Langlands correspondences.
Let $K$ be a complete discretely valued field of mixed characteristic $(0, p)$ with perfect residue field. We prove that the category of prismatic $F$-crystals on $\mathcal{O}_K$ is equivalent to the category of lattices in crystalline $G_K$-representations.