[Paper Review] Measure and sliding stability for 2-dimensional minimal cones in Euclidean spaces

Xiangyu Liang|arXiv (Cornell University)|Aug 29, 2018

Geometric Analysis and Curvature Flows13 references3 citations

TL;DR

This paper establishes measure stability for all 2-dimensional Almgren minimal cones in $\mathbb{R}^n$ and proves Almgren and topological sliding stability for all such cones in $\mathbb{R}^3$. The key contribution is that these stability properties, combined with uniqueness, allow the construction of new minimal cones via almost orthogonal unions, significantly expanding the known families of minimal sets in geometric measure theory.

ABSTRACT

In this article we prove the measure stability for all 2-dimensional Almgren minimal cones in $\mathbb{R}^n$, and the Almgren (resp. topological) sliding stability for the 2-dimensional Almgren (resp. topological) minimal cones in $\mathbb{R}^3$. As proved in \cite{2T}, when several 2-dimensional Almgren (resp. topological) minimal cones are measure and Almgren (resp. topological) sliding stable, and Almgren (resp. topological) unique, the almost orthogonal union of them stays minimal. As consequence, the results of this article, together with the uniqueness properties proved in \cite{uniquePYT}, permit us to use all 2-dimensional minimal cones in $\mathbb{R}^3$ to generate new families of minimal cones by taking their almost orthogonal unions.

Motivation & Objective

• To establish measure stability for all 2-dimensional Almgren minimal cones in $\mathbb{R}^n$.
• To prove Almgren and topological sliding stability for all 2-dimensional minimal cones in $\mathbb{R}^3$.
• To provide foundational stability results that, when combined with uniqueness, allow the generation of new minimal cones through almost orthogonal unions.

Proposed method

• Second-order estimates of Hausdorff measure under center shifts and boundary sliding are used to prove measure stability.
• Separation and connectedness conditions are applied repeatedly to control measure in sliding stability proofs.
• The paired calibration method is combined with geometric constraints to analyze competitors.
• Projection-based analysis on symmetric domains (e.g., four 3D sectors) is used to compare measure changes across competitors.
• Symmetry and invariance under permutations of indices are exploited to equate measure sums across different configurations.
• The proof leverages known results from [13] and [15], particularly the role of uniqueness and stability in preserving minimality under unions.

Experimental results

Research questions

• RQ1 Are all 2-dimensional Almgren minimal cones in $\mathbb{R}^n$ measure stable?
• RQ2 Do all 2-dimensional Almgren minimal cones in $\mathbb{R}^3$ satisfy Almgren sliding stability?
• RQ3 Do all 2-dimensional topological minimal cones in $\mathbb{R}^3$ satisfy topological sliding stability?
• RQ4 Can almost orthogonal unions of stable, unique 2-dimensional minimal cones remain minimal?
• RQ5 What geometric and measure-theoretic conditions ensure that a union of minimal cones is itself minimal?

Key findings

• All 2-dimensional Almgren minimal cones in $\mathbb{R}^n$ are measure stable, as established in Theorem 3.1.
• All 2-dimensional Almgren minimal cones in $\mathbb{R}^3$ are Almgren sliding stable, as proven in Theorem 5.11.
• All 2-dimensional topological minimal cones in $\mathbb{R}^3$ are topological sliding stable, as confirmed in Theorem 5.1 and 5.2.
• The $\mathbb{T}$ set is $(\eta, R_1(\eta))$-Almgren sliding stable for all $\eta < \frac{1}{2}$ with $R_1(\eta) = \sqrt{1 - (1 - \eta)^2}$, as shown in Corollary 5.12.
• The almost orthogonal union of measure and sliding stable, uniquely defined 2-dimensional minimal cones remains minimal, extending the known families of minimal sets.