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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.23074 |
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| _version_ | 1866911292122464256 |
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| author | Li, Xiang-Dong Yan, Qi |
| author_facet | Li, Xiang-Dong Yan, Qi |
| contents | In this paper, we study Perelman' s $ \mathcal{W}$ entropy for mean curvature flow in $\mathbb{R}^{n+1}$. Analogously to Perelman's $\mathcal{W}$-entropy defined for Ricci flow, K. Ecker in \cite{Ecker07} defined a functional $\mathcal{W}$ for the mean curvature flow in $\mathbb{R}^{n+1}$ and the region it encloses, and made the conjecture that this functional is monotonically increasing in time. We modify K. Ecker's definition and, using Hamilton's Harnack inequality for mean curvature flow, prove that our redefined $\mathcal{W}$-entropy is monotonically decreasing in time. Additionally, we provide a rigidity theorem for this $\mathcal{W}$-entropy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monotonicity of Perelman $\mathcal{W}$-Entropy of Mean Curvature Flow Li, Xiang-Dong Yan, Qi Differential Geometry 53E10 In this paper, we study Perelman' s $ \mathcal{W}$ entropy for mean curvature flow in $\mathbb{R}^{n+1}$. Analogously to Perelman's $\mathcal{W}$-entropy defined for Ricci flow, K. Ecker in \cite{Ecker07} defined a functional $\mathcal{W}$ for the mean curvature flow in $\mathbb{R}^{n+1}$ and the region it encloses, and made the conjecture that this functional is monotonically increasing in time. We modify K. Ecker's definition and, using Hamilton's Harnack inequality for mean curvature flow, prove that our redefined $\mathcal{W}$-entropy is monotonically decreasing in time. Additionally, we provide a rigidity theorem for this $\mathcal{W}$-entropy. |
| title | Monotonicity of Perelman $\mathcal{W}$-Entropy of Mean Curvature Flow |
| topic | Differential Geometry 53E10 |
| url | https://arxiv.org/abs/2511.23074 |