Nash entropy, Calabi energy and geometric regularization of singular Kähler metrics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912218851835904 |
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| author | Guo, Bin Song, Jian |
| author_facet | Guo, Bin Song, Jian |
| contents | We prove uniform Sobolev bounds for solutions of the Laplace equation on a general family of Kähler manifolds with bounded Nash entropy and Calabi energy. These estimates establish a connection to the theory of RCD spaces and provide abundant examples of RCD spaces topologically and holomorphically equivalent to projective varieties. Suppose $X$ is a normal projective variety that admits a resolution of singularities with relative nef or relative effective anti-canonical bundle. Then every admissible singular Kähler metric on $X$ with Ricci curvature bounded below induces a non-collapsed RCD space homeomorphic to the projective variety $X$ itself. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nash entropy, Calabi energy and geometric regularization of singular Kähler metrics Guo, Bin Song, Jian Differential Geometry Analysis of PDEs Metric Geometry We prove uniform Sobolev bounds for solutions of the Laplace equation on a general family of Kähler manifolds with bounded Nash entropy and Calabi energy. These estimates establish a connection to the theory of RCD spaces and provide abundant examples of RCD spaces topologically and holomorphically equivalent to projective varieties. Suppose $X$ is a normal projective variety that admits a resolution of singularities with relative nef or relative effective anti-canonical bundle. Then every admissible singular Kähler metric on $X$ with Ricci curvature bounded below induces a non-collapsed RCD space homeomorphic to the projective variety $X$ itself. |
| title | Nash entropy, Calabi energy and geometric regularization of singular Kähler metrics |
| topic | Differential Geometry Analysis of PDEs Metric Geometry |
| url | https://arxiv.org/abs/2502.02041 |