Eigenvalue Estimate for the Rough Laplacian on $1$-Forms and its Applications
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| Format: | Preprint |
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2025
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| _version_ | 1866918231519789056 |
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| author | Huang, Teng Wang, Weiwei |
| author_facet | Huang, Teng Wang, Weiwei |
| contents | In this article, we establish a geometric lower bound for the first positive eigenvalue $λ^{(1)}_{1}$ of the rough Laplacian acting on $1$-forms for closed $2n$-dimensional Riemannian manifolds with nonvanishing Euler characteristic. In contrast to the case of functions, such a Li-Yau-type estimate does not hold in general, as evidenced by existing counterexamples. Under assumptions including a lower bound on Ricci curvature, an upper bound on diameter, and an $L^{2p}$-norm bound on the Riemann curvature tensor, we prove that $λ^{(1)}_{1}$ is bounded below by a positive constant depending on these parameters. As applications, we derive vanishing results for the Euler characteristic under certain Ricci curvature bounds and the presence of a nonzero Killing vector field, extending classical Bochner-type theorems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_04740 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenvalue Estimate for the Rough Laplacian on $1$-Forms and its Applications Huang, Teng Wang, Weiwei Differential Geometry In this article, we establish a geometric lower bound for the first positive eigenvalue $λ^{(1)}_{1}$ of the rough Laplacian acting on $1$-forms for closed $2n$-dimensional Riemannian manifolds with nonvanishing Euler characteristic. In contrast to the case of functions, such a Li-Yau-type estimate does not hold in general, as evidenced by existing counterexamples. Under assumptions including a lower bound on Ricci curvature, an upper bound on diameter, and an $L^{2p}$-norm bound on the Riemann curvature tensor, we prove that $λ^{(1)}_{1}$ is bounded below by a positive constant depending on these parameters. As applications, we derive vanishing results for the Euler characteristic under certain Ricci curvature bounds and the presence of a nonzero Killing vector field, extending classical Bochner-type theorems. |
| title | Eigenvalue Estimate for the Rough Laplacian on $1$-Forms and its Applications |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2512.04740 |