On EulerDierkesHuisken 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 areafunctional $$\mathcal{E}_f(M)=\int_M f(x)\; d \mathcal{H}_k$$ with the density function $f(x)=g(x)$ and $g(t)$ is nonnegative, 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 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 $x^\alpha$stable for $\alpha\geqk+2\sqrt{2(k1)}$, the oriented stable minimal hypercones are $x^\alpha$stable for $\alpha\geq 0$, 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 $x^\alpha$minimizing for $\dim \mathcal{C} \geq 7$, $k_i>1$ and $\alpha \geq 0$, the Simons cones $C(S^{p} \times S^{p})(p\geq 1)$ are $x^\alpha$minimizing for any $\alpha \geq 1$, which relaxes the assumption $1\leq\alpha \leq 2p$ in \cite{DH23}.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.05132
 arXiv:
 arXiv:2404.05132
 Bibcode:
 2024arXiv240405132C
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs