Strict Stability and Strict Minimality of Regular Area-Minimizing Hypercones: A Quantitative Characterization
Let $\mathbf{C}=\partial E\subset \mathbb{R}^{n+1}$ be a regular area-minimizing hypercone. We first prove that $\mathbf{C}$ is simultaneously strictly stable and strictly minimizing if and only if there exists $c_\mathbf{C}>0$ such that \[ \operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R) \geq c_\mathbf{C} \int_{F\mathbin\triangle E} \frac{\operatorname{dist}(x,\mathbf{C})}{|x|^2}\,dx \] for every $R>0$ and finite-perimeter set $F$ with $F\mathbin\triangle E\Subset B_R$. Thus this intrinsic distance-weighted inequality characterizes exactly the simultaneous strictness of the stability and minimizing properties. Second, without either strictness assumption, every regular area-minimizing hypercone satisfies the scale-invariant quadratic inequality \[ \frac{\operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R)}{R^n} \geq c_\mathbf{C} \left( \frac{|F\mathbin\triangle E|}{R^{n+1}} \right)^2. \] This extends the inequality previously established for area-minimizing Lawson cones. Finally, the area-minimizing assumption is unnecessary for our spectral result: for every stable regular minimal hypercone, the first Dirichlet eigenvalue $\lambda_\mathbf{C}^D(R)$ of the Jacobi operator on $\mathbf{C}\cap B_R$ is given exactly by \[ \lambda_\mathbf{C}^D(R) = \frac{j_{b_1,1}^2}{R^2}, \qquad b_1^2 = \frac{(n-2)^2}{4}+\mu_1, \] where $\mu_1$ is the first eigenvalue of the link Jacobi operator and $j_{b_1,1}$ is the first positive zero of the Bessel function $J_{b_1}$. In particular, this identifies the optimal Dirichlet spectral constant for every stable regular minimal hypercone.