On the stability of the equator map for higher order energy functionals
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866909451747852288 |
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| author | Fardoun, Ali Montaldo, Stefano Ratto, Andrea |
| author_facet | Fardoun, Ali Montaldo, Stefano Ratto, Andrea |
| contents | Let $B^n\subset {\mathbb R}^{n}$ and ${\mathbb S}^n\subset {\mathbb R}^{n+1}$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. The \textit{extrinsic $k$-energy functional} is defined on the Sobolev space $W^{k,2}\left (B^n,{\mathbb S}^n \right )$ as follows: $E_{k}^{\rm ext}(u)=\int_{B^n}|Δ^s u|^2\,dx$ when $k=2s$, and $E_{k}^{\rm ext}(u)=\int_{B^n}|\nabla Δ^s u|^2\,dx$ when $k=2s+1$. These energy functionals are a natural higher order version of the classical extrinsic bienergy, also called Hessian energy. The equator map $u^*: B^n \to {\mathbb S}^n$, defined by $u^*(x)=(x/|x|,0)$, is a critical point of $E_{k}^{\rm ext}(u)$ provided that $n \geq 2k+1$. The main aim of this paper is to establish necessary and sufficient conditions on $k$ and $n$ under which $u^*: B^n \to {\mathbb S}^n$ is minimizing or unstable for the extrinsic $k$-energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_01509 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the stability of the equator map for higher order energy functionals Fardoun, Ali Montaldo, Stefano Ratto, Andrea Differential Geometry Let $B^n\subset {\mathbb R}^{n}$ and ${\mathbb S}^n\subset {\mathbb R}^{n+1}$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. The \textit{extrinsic $k$-energy functional} is defined on the Sobolev space $W^{k,2}\left (B^n,{\mathbb S}^n \right )$ as follows: $E_{k}^{\rm ext}(u)=\int_{B^n}|Δ^s u|^2\,dx$ when $k=2s$, and $E_{k}^{\rm ext}(u)=\int_{B^n}|\nabla Δ^s u|^2\,dx$ when $k=2s+1$. These energy functionals are a natural higher order version of the classical extrinsic bienergy, also called Hessian energy. The equator map $u^*: B^n \to {\mathbb S}^n$, defined by $u^*(x)=(x/|x|,0)$, is a critical point of $E_{k}^{\rm ext}(u)$ provided that $n \geq 2k+1$. The main aim of this paper is to establish necessary and sufficient conditions on $k$ and $n$ under which $u^*: B^n \to {\mathbb S}^n$ is minimizing or unstable for the extrinsic $k$-energy. |
| title | On the stability of the equator map for higher order energy functionals |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2007.01509 |