Liouville theorem for biharmonic functions on manifolds of nonnegative Ricci curvature
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908683532763136 |
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| author | Bravo, John E. Cortissoz, Jean C. |
| author_facet | Bravo, John E. Cortissoz, Jean C. |
| contents | In this paper we extend Yau's celebrated Liouville theorem to the biharmonic case. Namely, we show that in a complete Riemannian manifold with a pole and nonnegative Ricci curvature, any biharmonic function of subquadratic growth must be harmonic, and hence, any biharmonic function of sublinear growth must be constant. Our proof relies on a new local $L^2$ estimate for the Laplacian of biharmonic functions combined with a mean value inequality. Examples where our theorem applies include hypersurfaces of positive sectional curvature in $\mathbb{R}^n$, and manifolds with a pole of nonnegative Ricci curvature whose curvature decays at infinity rapidly enough. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouville theorem for biharmonic functions on manifolds of nonnegative Ricci curvature Bravo, John E. Cortissoz, Jean C. Differential Geometry In this paper we extend Yau's celebrated Liouville theorem to the biharmonic case. Namely, we show that in a complete Riemannian manifold with a pole and nonnegative Ricci curvature, any biharmonic function of subquadratic growth must be harmonic, and hence, any biharmonic function of sublinear growth must be constant. Our proof relies on a new local $L^2$ estimate for the Laplacian of biharmonic functions combined with a mean value inequality. Examples where our theorem applies include hypersurfaces of positive sectional curvature in $\mathbb{R}^n$, and manifolds with a pole of nonnegative Ricci curvature whose curvature decays at infinity rapidly enough. |
| title | Liouville theorem for biharmonic functions on manifolds of nonnegative Ricci curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2511.08358 |