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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2512.07389 |
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| _version_ | 1866918237732601856 |
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| author | Lincastri, Salvatore |
| author_facet | Lincastri, Salvatore |
| contents | In this paper, we discuss the validity of the Liouville property for $X$-harmonic functions, i.e. positive solution to $Δ_{X}u=0$, where $X$ is a vector field on a complete, non-compact Riemannian manifold and $Δ_{X}$ is the drifted Laplacian. In particular, we show that if the $X$-Bakry-Émery-Ricci curvature $\mathrm{Ric}_{X}$ is non-negative and the norm of $X$ decays to zero at infinity, then the manifold has the Liouville property for the $X$-Laplacian. The proof exploits a local gradient estimate for positive solutions to the semilinear equation $Δ_{X}u+F(u)=0$, which holds when $F$ satisfies the structural conditions $tF'(t)-F(t)\leα$ and $\vert F(t)\vert\leβt$, and the manifold has $\mathrm{Ric}_{X}\ge-(n-1)K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07389 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gradient estimates and Liouville properties for the drifted Laplacian Lincastri, Salvatore Differential Geometry Analysis of PDEs In this paper, we discuss the validity of the Liouville property for $X$-harmonic functions, i.e. positive solution to $Δ_{X}u=0$, where $X$ is a vector field on a complete, non-compact Riemannian manifold and $Δ_{X}$ is the drifted Laplacian. In particular, we show that if the $X$-Bakry-Émery-Ricci curvature $\mathrm{Ric}_{X}$ is non-negative and the norm of $X$ decays to zero at infinity, then the manifold has the Liouville property for the $X$-Laplacian. The proof exploits a local gradient estimate for positive solutions to the semilinear equation $Δ_{X}u+F(u)=0$, which holds when $F$ satisfies the structural conditions $tF'(t)-F(t)\leα$ and $\vert F(t)\vert\leβt$, and the manifold has $\mathrm{Ric}_{X}\ge-(n-1)K$. |
| title | Gradient estimates and Liouville properties for the drifted Laplacian |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2512.07389 |