Zero modes on product Riemannian manifolds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908909791346688 |
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| author | Julio-Batalla, Jurgen |
| author_facet | Julio-Batalla, Jurgen |
| contents | This paper is concerned with the zero mode equation $D_gφ=iA\cdotφ$ on product of closed spin manifolds $(M_1^{n_1}\times M_2^{n_2},g_1+g_2,σ)$ of dimensions $n_1\leq n_2$ respectively. Here $A$ is a real vector field on $M^n=M_1^{n_1}\times M_2^{n_2}$. Under non-increasing condition on $|φ|$ we prove that $$\parallel A\parallel_n^2\geq\frac{n_2}{4(n_2-1)}Y(M^n,[g]),$$ where $Y(M^n,[g])$ is the Yamabe constant of $(M^n,g)$. This estimate is sharp in even dimensions. We also obtain a similar estimate for non trivial solutions of the zero mode type equation $D_gφ=fφ$, where $f$ is a scalar function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23242 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Zero modes on product Riemannian manifolds Julio-Batalla, Jurgen Differential Geometry Spectral Theory This paper is concerned with the zero mode equation $D_gφ=iA\cdotφ$ on product of closed spin manifolds $(M_1^{n_1}\times M_2^{n_2},g_1+g_2,σ)$ of dimensions $n_1\leq n_2$ respectively. Here $A$ is a real vector field on $M^n=M_1^{n_1}\times M_2^{n_2}$. Under non-increasing condition on $|φ|$ we prove that $$\parallel A\parallel_n^2\geq\frac{n_2}{4(n_2-1)}Y(M^n,[g]),$$ where $Y(M^n,[g])$ is the Yamabe constant of $(M^n,g)$. This estimate is sharp in even dimensions. We also obtain a similar estimate for non trivial solutions of the zero mode type equation $D_gφ=fφ$, where $f$ is a scalar function. |
| title | Zero modes on product Riemannian manifolds |
| topic | Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2603.23242 |