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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.15236 |
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| _version_ | 1866911073139949568 |
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| author | Honda, Shouhei Kristály, Alexandru Pîrvuceanu, Alexandru |
| author_facet | Honda, Shouhei Kristály, Alexandru Pîrvuceanu, Alexandru |
| contents | The main goal of the present paper is to provide sharp hypercontractivity bounds of the heat flow $({\sf H}_t)_{t\geq 0}$ on ${\sf RCD}(0,N)$ metric measure spaces. The best constant in this estimate involves the asymptotic volume ratio, and its optimality is obtained by means of the sharp $L^2$-logarithmic Sobolev inequality on ${\sf RCD}(0,N)$ spaces and a blow-down rescaling argument. Equality holds in this sharp estimate for a prescribed time $t_0>0$ and a non-zero extremizer $f$ if and only if the ${\sf RCD}(0,N)$ space has an $N$-Euclidean cone structure and $f$ is a Gaussian whose dilation factor is reciprocal to $t_0$, up to a multiplicative constant. Applications include an extension of Li's rigidity result, almost rigidities, as well as topological rigidities of non-collapsed ${\sf RCD}(0, N)$ spaces. Our results are new even on complete Riemannian manifolds with non-negative Ricci curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15236 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hypercontractivity of the heat flow on ${\sf RCD}(0,N)$ spaces: sharpness and rigidities Honda, Shouhei Kristály, Alexandru Pîrvuceanu, Alexandru Analysis of PDEs Functional Analysis The main goal of the present paper is to provide sharp hypercontractivity bounds of the heat flow $({\sf H}_t)_{t\geq 0}$ on ${\sf RCD}(0,N)$ metric measure spaces. The best constant in this estimate involves the asymptotic volume ratio, and its optimality is obtained by means of the sharp $L^2$-logarithmic Sobolev inequality on ${\sf RCD}(0,N)$ spaces and a blow-down rescaling argument. Equality holds in this sharp estimate for a prescribed time $t_0>0$ and a non-zero extremizer $f$ if and only if the ${\sf RCD}(0,N)$ space has an $N$-Euclidean cone structure and $f$ is a Gaussian whose dilation factor is reciprocal to $t_0$, up to a multiplicative constant. Applications include an extension of Li's rigidity result, almost rigidities, as well as topological rigidities of non-collapsed ${\sf RCD}(0, N)$ spaces. Our results are new even on complete Riemannian manifolds with non-negative Ricci curvature. |
| title | Hypercontractivity of the heat flow on ${\sf RCD}(0,N)$ spaces: sharpness and rigidities |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2503.15236 |