Sharp gradient estimates and monotonicity in positive Ricci curvature
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915656313602048 |
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| author | Manea, Cosmin |
| author_facet | Manea, Cosmin |
| contents | We prove a sharp gradient estimate for the natural Green's function of a closed manifold with positive Ricci curvature. We also show that this estimate is closely related to a family of monotonicity formulae. These results extend those previously obtained by Colding and Minicozzi for open manifolds with non-negative Ricci curvature. We further obtain several geometric applications, including a new proof of Bishop's volume comparison theorem in dimension four. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05467 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp gradient estimates and monotonicity in positive Ricci curvature Manea, Cosmin Differential Geometry Analysis of PDEs We prove a sharp gradient estimate for the natural Green's function of a closed manifold with positive Ricci curvature. We also show that this estimate is closely related to a family of monotonicity formulae. These results extend those previously obtained by Colding and Minicozzi for open manifolds with non-negative Ricci curvature. We further obtain several geometric applications, including a new proof of Bishop's volume comparison theorem in dimension four. |
| title | Sharp gradient estimates and monotonicity in positive Ricci curvature |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2512.05467 |