On Euler-Dierkes-Huisken variational problem

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 $f(x)=g(\mathrm{\mid }x\mathrm{\mid })$ 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 $\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }$. 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 $\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }$-stable for $\alpha \ge -k+2\sqrt{2(k-1)}$, the oriented stable minimal hypercones are $\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }$-stable for $\alpha \ge 0$, and we also show that the minimal cones over product of spheres $\mathcal{C}=C(S^{k_1}\times \cdots \times S^{k_m})$ are $\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }$-minimizing for $\dim \mathcal{C}\ge 7$, $k_i>1$ and $\alpha \ge 0$, the Simons cones $C(S^p\times S^p)(p\ge 1)$ are $\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }$-minimizing for any $\alpha \ge 1$, which relaxes the assumption $1\le \alpha \le 2p$ in \cite{DH23}.