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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.01970 |
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| _version_ | 1866912503466819584 |
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| author | Shen, Bin |
| author_facet | Shen, Bin |
| contents | This manuscript investigates the curvature and topological properties of certain $\infty$-Einstein Finsler metrics on Finsler metric measure spaces. By imposing symmetry conditions, we construct a series of special metrics and analyze their equivalence on special manifolds. Provided a Ricci curvature bound, we establish a linear growth lower bound estimate for the S-curvature and the distortion, revealing the interplay between curvature and measure on $\infty$-Einstein Finsler manifolds. Furthermore, by introducing scalar curvature and imposing a linear growth lower bound condition, we derive upper and lower bounds for the distortion, S-curvature, and the scalar curvature itself on asymmetric essential gradient Ricci solitons with certain non-Riemannian curvature constraints. These results yield direct topological finiteness conclusions for some forward-complete $\infty$-Einstein Finsler manifolds. Our work partially addresses Gromov's conjecture of scalar curvature in the context of Finsler metric measure spaces and provides a foundation for further research in geometric analysis within general Finsler geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01970 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds of Scalar curvature, S-curvature and distortion on $\infty$-Einstein Finsler manifolds Shen, Bin Differential Geometry This manuscript investigates the curvature and topological properties of certain $\infty$-Einstein Finsler metrics on Finsler metric measure spaces. By imposing symmetry conditions, we construct a series of special metrics and analyze their equivalence on special manifolds. Provided a Ricci curvature bound, we establish a linear growth lower bound estimate for the S-curvature and the distortion, revealing the interplay between curvature and measure on $\infty$-Einstein Finsler manifolds. Furthermore, by introducing scalar curvature and imposing a linear growth lower bound condition, we derive upper and lower bounds for the distortion, S-curvature, and the scalar curvature itself on asymmetric essential gradient Ricci solitons with certain non-Riemannian curvature constraints. These results yield direct topological finiteness conclusions for some forward-complete $\infty$-Einstein Finsler manifolds. Our work partially addresses Gromov's conjecture of scalar curvature in the context of Finsler metric measure spaces and provides a foundation for further research in geometric analysis within general Finsler geometry. |
| title | Bounds of Scalar curvature, S-curvature and distortion on $\infty$-Einstein Finsler manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2501.01970 |