On Euler-Dierkes-Huisken variational problem

Abstract

In this paper, we study the stability and minimizing properties of higher codimensional surfaces in Euclidean space associated with the $f$-weighted area-functional $$\mathcal{E}_f(M)=\int_M f(x)\; d \mathcal{H}_k$$ with the density function $f(x)=g(|x|)$ and $g(t)$ is non-negative, which develop the recent works by U. Dierkes and G. Huisken (Math. Ann., 20 October 2023) on hypersurfaces with the density function $|x|^α$. Under suitable assumptions on $g(t)$, we prove that minimal cones with globally flat normal bundles are $f$-stable, and we also prove that the regular minimal cones satisfying Lawlor curvature criterion, the highly singular determinantal varieties and Pfaffian varieties without some exceptional cases are $f$-minimizing. As an application, we show that $k$-dimensional minimal cones over product of spheres are $|x|^α$-stable for $α\geq-k+2\sqrt{2(k-1)}$, the oriented stable minimal hypercones are $|x|^α$-stable for $α\geq 0$, and we also show that the minimal cones over product of spheres $\mathcal{C}=C \left(S^{k_1} \times \cdots \times S^{k_{m}}\right)$ are $|x|^α$-minimizing for $\dim \mathcal{C} \geq 7$, $k_i>1$ and $α\geq 0$, the Simons cones $C(S^{p} \times S^{p})(p\geq 1)$ are $|x|^α$-minimizing for any $α\geq 1$, which relaxes the assumption $1\leqα\leq 2p$ in \cite{DH23}.