Nash entropy, Calabi energy and geometric regularization of singular K\"ahler metrics

Bin Guo; Jian Song

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arXiv · 2502.02041

Nash entropy, Calabi energy and geometric regularization of singular K\"ahler metrics

Bin Guo·Jian Song

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Abstract

We prove uniform Sobolev bounds for solutions of the Laplace equation on a general family of K\"ahler 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 K\"ahler metric on $X$ with Ricci curvature bounded below induces a non-collapsed RCD space homeomorphic to the projective variety $X$ itself.

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Bin Guo, Jian Song. 2025-02-04. Nash entropy, Calabi energy and geometric regularization of singular K\"ahler metrics. https://arxiv.org/abs/2502.02041

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