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 E-f (M) = integral(M) f (x) dH(k) with the density function f (x) = g(vertical bar x vertical bar) 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 vertical bar x vertical bar a. 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 vertical bar x vertical bar(alpha)-stable for alpha >= -k + 2 root 2(k - 1), the oriented stable minimal hypercones are vertical bar x vertical bar(alpha)-stable for alpha >= 0, and we also show that the cones over product of spheres C = C (S-k1 x center dot center dot center dot x S-km) are vertical bar x vertical bar(alpha)-minimizing for dim C >= 7, k(i) > 1 and alpha >= 0, the Simons cones C(S-p x S-p) are vertical bar x vertical bar alpha(a)-minimizing for alpha >= 1, which relaxes the assumption 1 <= alpha <= 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 x S-p) are vertical bar x vertical bar(alpha)-minimizing for alpha >= 3 - p, which has improved our assumption alpha >= 1 for p >= 3.