[Paper Review] Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s\sim 1$

Xavier Cabré, Eleonora Cinti+1 more|arXiv (Cornell University)|Oct 24, 2017

Nonlinear Partial Differential Equations11 references3 citations

TL;DR

This paper proves that in $ℝ^3$, the only stable $s$-minimal cones for $s$ sufficiently close to 1 are half-spaces, establishing the first classification result for stable nonlocal minimal cones in dimension higher than two. The proof relies on a geometric second variation formula involving a nonlocal second fundamental form and sharp BV estimates, avoiding compactness arguments from $s=1$ by providing an explicit quantitative bound on $s$ near 1.

ABSTRACT

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 can not 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.

Motivation & Objective

• To classify stable $s$-minimal cones in $ℝ^3$ for $s$ close to 1, extending known results from $s=1$ and $n=2$ to higher dimensions.
• To overcome the failure of compactness arguments in the nonlocal setting, where sequences of stable cones for $s_k \uparrow 1$ may not converge due to unbounded perimeter growth.
• To establish a quantitative threshold for $s$ near 1 such that only half-spaces are stable, providing a non-perturbative proof independent of the classical case.
• To apply recent sharp BV and energy estimates for stable nonlocal minimal sets to control the geometry of the boundary of $s$-minimal cones.

Proposed method

• Use the geometric second variation formula for the fractional $s$-perimeter, which involves a squared nonlocal second fundamental form, to analyze stability of $s$-minimal cones.
• Apply sharp $BV$ and energy estimates for stable nonlocal minimal sets to control the total variation and regularity of the boundary of $\Sigma$.
• Analyze the trace of $\partial\Sigma$ on the unit sphere $S^2$, decomposing it into finitely many $C^1$ curves $\gamma_i$ with lengths $L_i$, and use the second variation to derive a key inequality involving the normal vector differences and the kernel $k_s(\hat{x},\hat{y}) = |\hat{x}-\hat{y}|^{-2-s}$.
• Use a contradiction argument assuming more than one curve in the trace: if $J>1$, then the sum of weighted integrals of $|\nu_\Sigma(\hat{x}) - \nu_\Sigma(\hat{y})|^2 k_s(\hat{x},\hat{y})$ over pairs of curves leads to a lower bound that contradicts the $C^1$-regularity and $C^{1/4}$-closeness to great circles when $s$ is close to 1.
• Employ a foliation argument and viscosity solution theory to show that if the boundary is a Lipschitz graph, then it must be smooth and hence a plane, implying $\Sigma$ is a half-space.
• Use a refined $L^2$-type estimate and the nonlocal kernel to derive a contradiction when multiple curves are present, especially when they are close to great circles with opposite or same orientation.

Experimental results

Research questions

• RQ1 Are half-spaces the only stable $s$-minimal cones in $\mathbb{R}^3$ for $s$ sufficiently close to 1, beyond the classical $s=1$ case?
• RQ2 Can the classification of stable $s$-minimal cones in $\mathbb{R}^3$ be achieved without relying on compactness arguments from the $s=1$ limit?
• RQ3 What is the quantitative threshold for $s$ near 1 that guarantees flatness of stable $s$-minimal cones in $\mathbb{R}^3$?
• RQ4 How does the nonlocal second variation formula, involving a nonlocal second fundamental form, constrain the geometry of $s$-minimal cones?
• RQ5 Can sharp $BV$ estimates for stable nonlocal minimal sets be used to rule out non-flat configurations in $\mathbb{R}^3$?

Key findings

• The only stable $s$-minimal cones in $\mathbb{R}^3$ are half-spaces when $s$ is sufficiently close to 1, establishing the first classification result in dimension $n > 2$.
• The required closeness of $s$ to 1 is quantifiable: the proof yields a universal constant $C$ such that if $s > 1 - \delta$ for some $\delta > 0$ depending on $C$, then only half-spaces are stable.
• The proof does not rely on compactness arguments from $s=1$, as sequences of stable cones for $s_k \uparrow 1$ may fail to be precompact due to unbounded perimeter growth on $S^2$.
• The second variation formula for the $s$-perimeter involves a squared nonlocal second fundamental form, which is central to the analysis and leads to a contradiction when multiple curves are present in the spherical trace.
• If the boundary of the cone is a Lipschitz graph, then it is $C^\infty$ and hence a hyperplane, implying the cone is a half-space.
• A contradiction arises when assuming more than one curve in the spherical trace: either curves with opposite orientation or same orientation with a third curve in between lead to a lower bound on the second variation that violates the stability inequality for $s$ close to 1.