Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $ε$-range and it's application
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916757353005056 |
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| author | Li, Wen-Qi Zhang, Zhikai |
| author_facet | Li, Wen-Qi Zhang, Zhikai |
| contents | In this paper, we establish a parabolic Harnack inequality for positive solutions of the $ϕ$-heat equation and prove Gaussian upper and lower bounds for the $ϕ$-heat kernel on weighted Riemannian manifolds under lower $N$-Ricci curvature bound with $\varepsilon$-range. Building on these results, we demonstrate: The $L^1_ϕ$-Liouville theorem for $ϕ$-subharmonic functions, $L^1_ϕ$-uniqueness property for solutions of the $ϕ$-heat equation and lower bounds for eigenvalues of the weighted Laplacian $Δ_ϕ$.
Furthermore, leveraging the Gaussian upper bound of the weighted heat kernel, we construct a Li-Yau-type gradient estimate for the positive solution of weighted heat equation under a weighted $L^p(μ)$-norm constraint on $|\nablaϕ|^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19113 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $ε$-range and it's application Li, Wen-Qi Zhang, Zhikai Differential Geometry Analysis of PDEs In this paper, we establish a parabolic Harnack inequality for positive solutions of the $ϕ$-heat equation and prove Gaussian upper and lower bounds for the $ϕ$-heat kernel on weighted Riemannian manifolds under lower $N$-Ricci curvature bound with $\varepsilon$-range. Building on these results, we demonstrate: The $L^1_ϕ$-Liouville theorem for $ϕ$-subharmonic functions, $L^1_ϕ$-uniqueness property for solutions of the $ϕ$-heat equation and lower bounds for eigenvalues of the weighted Laplacian $Δ_ϕ$. Furthermore, leveraging the Gaussian upper bound of the weighted heat kernel, we construct a Li-Yau-type gradient estimate for the positive solution of weighted heat equation under a weighted $L^p(μ)$-norm constraint on $|\nablaϕ|^2$. |
| title | Heat kernel estimate on weighted Riemannian manifolds under lower $N$-Ricci curvature bounds with $ε$-range and it's application |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2505.19113 |