Translational and Sliding Stability for Two-Dimensional Minimal Cones in Euclidean Spaces

In this article, we prove the translational stability for all two-dimensional Almgren minimal cones in ${\mathbb{R}}^n$ and the Almgren (resp. topological) sliding stability for the two-dimensional Almgren (resp. topological) minimal cones in ${\mathbb{R}}^3$. As proved in [ 19], when several two-dimensional Almgren (resp. topological) minimal cones are translational, Almgren (resp. topological) sliding stable, and Almgren (resp. topological) unique, their almost orthogonal union stays minimal. As a consequence, the results of this article, together with the uniqueness properties proved in [ 14], permit us to use all two-dimensional minimal cones in ${\mathbb{R}}^3$ to generate new families of minimal cones by taking their almost orthogonal unions.

Stable s-minimal cones in ℝ3 are flat for s ~ 1

We prove that half spaces are the only stable nonlocal s-minimal cones in {\mathbb{R}^{3}}, for {s\in(0,1)} sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from {s=1}. In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

V.M. Miklyukov: from Dimension 8 to Nonassociative Algebras

In this short survey we give a background and explain some recent developments in algebraic minimal cones and nonassociative algebras. A part of this paper is recollections of my collaboration with my teacher, PhD supervisor and a colleague, Vladimir Miklyukov on minimal surface theory that motivated the present research.