On Perelman's $W$-entropy and Shannon entropy power for super Ricci flows on metric measure spaces
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910929523834880 |
|---|---|
| author | Li, Xiang-Dong |
| author_facet | Li, Xiang-Dong |
| contents | In this paper, we extend Perelman's $W$-entropy formula and the concavity of the Shannon entropy power from smooth Ricci flow to super Ricci flows on metric measure spaces. Moreover, we prove the Li-Yau-Hamilton-Perelman Harnack inequality on super Ricci flows. As a significant application, we prove the equivalence between the volume non-local collapsing property and the lower boundedness of the $W$-entropy on RCD$(0, N)$ spaces. Finally, we use the $W$-entropy to study the logarithmic Sobolev inequality with optimal constant on super Ricci flows on metric measure spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Perelman's $W$-entropy and Shannon entropy power for super Ricci flows on metric measure spaces Li, Xiang-Dong Differential Geometry Metric Geometry Probability In this paper, we extend Perelman's $W$-entropy formula and the concavity of the Shannon entropy power from smooth Ricci flow to super Ricci flows on metric measure spaces. Moreover, we prove the Li-Yau-Hamilton-Perelman Harnack inequality on super Ricci flows. As a significant application, we prove the equivalence between the volume non-local collapsing property and the lower boundedness of the $W$-entropy on RCD$(0, N)$ spaces. Finally, we use the $W$-entropy to study the logarithmic Sobolev inequality with optimal constant on super Ricci flows on metric measure spaces. |
| title | On Perelman's $W$-entropy and Shannon entropy power for super Ricci flows on metric measure spaces |
| topic | Differential Geometry Metric Geometry Probability |
| url | https://arxiv.org/abs/2505.03202 |