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

$$\begin{aligned} \mathcal {E}_f(M)=\int _M f(x)\; d \mathcal {H}_k \end{aligned}$$

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 \(|x|^\alpha \). 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 minimal cones satisfy the Lawlor curvature criterion, the determinantal varieties and the Pfaffian varieties without some exceptional cases are f-minimizing. As an application, we show that k-dimensional cones over product of spheres are \(|x|^\alpha \)-stable for \(\alpha \ge -k+2\sqrt{2(k-1)}\), the oriented stable minimal hypercones are \(|x|^\alpha \)-stable for \(\alpha \ge 0\), and we also show that the cones over product of spheres \(\mathcal {C}=C \left( S^{k_1} \times \cdots \times S^{k_{m}}\right) \) are \(|x|^\alpha \)-minimizing for \(\dim \mathcal {C} \ge 7\), \(k_i>1\) and \(\alpha \ge 0\), the Simons cones \(C(S^{p} \times S^{p})\) are \(|x|^\alpha \)-minimizing for \(\alpha \ge 1\), which relaxes the assumption \(1\le \alpha \le 2p\) in Dierkes and Huisken ( Math Ann, https://doi.org/10.1007/s00208-023-02726-3, 2023). Recently, Dierkes (Rend Sem Mat Univ Padova, 2024) prove that \(C(S^{p} \times S^{p})\) are \(|x|^\alpha \)-minimizing for \(\alpha \ge 3-p\), which has improved our assumption \(\alpha \ge 1\) for \(p\ge 3\).