Stability of spinorial Sobolev inequalities on $\mathbb{S}^n$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912545479065600 |
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| author | Wang, Guofang Zhang, Mingwei |
| author_facet | Wang, Guofang Zhang, Mingwei |
| contents | The spinorial Sobolev inequality on the unit sphere states \begin{equation*}
\Big(\int| Dψ|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}-\frac{n}{2}ω_{n}^{1/n}\int\langle Dψ,ψ\rangle
\geq 0, \end{equation*} with equality if and only if $ψ\in {\mathcal M}$, the set of all $-\frac 12$-Killing spinors and their conformal transformations.
Our main result in this paper is to refine this inequality by establishing a stability inequality
\begin{equation*} \Big(\int| Dψ|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}-\frac{n}{2}ω_{n}^{1/n}\int\langle Dψ,ψ\rangle
\geq {\bf c}_S\inf_{ϕ\in\mathcal{M}}\Big(\int| D(ψ-ϕ)|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}.
\end{equation*}
As a by-product of our argument, we show that elements in set $\mathcal M$ are not optimizers of another spinorial Sobolev inequality
\begin{equation*}
\Big(\int| Dψ|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}} \geq C_S \Big(\int|ψ|^{\frac{2n}{n-1}}\Big)^{\frac{n-1}{n}},
\end{equation*}
unlike expected by experts. They have in fact index $n+1$ and nullity $2^{[\frac n2]+2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability of spinorial Sobolev inequalities on $\mathbb{S}^n$ Wang, Guofang Zhang, Mingwei Differential Geometry Mathematical Physics Analysis of PDEs 53C27, 58C40, 35A23 The spinorial Sobolev inequality on the unit sphere states \begin{equation*} \Big(\int| Dψ|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}-\frac{n}{2}ω_{n}^{1/n}\int\langle Dψ,ψ\rangle \geq 0, \end{equation*} with equality if and only if $ψ\in {\mathcal M}$, the set of all $-\frac 12$-Killing spinors and their conformal transformations. Our main result in this paper is to refine this inequality by establishing a stability inequality \begin{equation*} \Big(\int| Dψ|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}-\frac{n}{2}ω_{n}^{1/n}\int\langle Dψ,ψ\rangle \geq {\bf c}_S\inf_{ϕ\in\mathcal{M}}\Big(\int| D(ψ-ϕ)|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}. \end{equation*} As a by-product of our argument, we show that elements in set $\mathcal M$ are not optimizers of another spinorial Sobolev inequality \begin{equation*} \Big(\int| Dψ|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}} \geq C_S \Big(\int|ψ|^{\frac{2n}{n-1}}\Big)^{\frac{n-1}{n}}, \end{equation*} unlike expected by experts. They have in fact index $n+1$ and nullity $2^{[\frac n2]+2}$. |
| title | Stability of spinorial Sobolev inequalities on $\mathbb{S}^n$ |
| topic | Differential Geometry Mathematical Physics Analysis of PDEs 53C27, 58C40, 35A23 |
| url | https://arxiv.org/abs/2508.09047 |