On Euler-Dierkes-Huisken variational problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915356601221120 |
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| author | Cui, Hongbin Xu, Xiaowei |
| author_facet | Cui, Hongbin Xu, Xiaowei |
| contents | 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}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_05132 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Euler-Dierkes-Huisken variational problem Cui, Hongbin Xu, Xiaowei Differential Geometry Analysis of PDEs 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}. |
| title | On Euler-Dierkes-Huisken variational problem |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2404.05132 |