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  "body": "\n## Summary\n\nThe Bergman metric of a bounded domain $\\Omega \\subseteq \\C^n$ is the Kähler metric whose potential is the logarithm of the [[Bergman kernel|8RWDHW0CXC]] on the diagonal.\nIt is defined from the Hilbert space of square-integrable holomorphic functions, the [[Bergman space|2D619WTPYC]].\nHence every biholomorphic map between bounded domains is an isometry for it.\nStefan Bergman introduced it, and on the unit disc it is the hyperbolic metric up to a constant factor.\nIt is a basic tool for the study of biholomorphic maps, for example in Fefferman's theorem on their boundary extension.\n\n## Explanations\n\n### L1 human\n\nA bounded domain has a natural Hilbert space of holomorphic functions, those whose squared modulus has finite integral.\nFrom this space one builds a kernel $K(z, w)$, and the logarithm of $K(z, z)$ serves as a potential for a Riemannian metric, the Bergman metric.\nBecause the construction uses only holomorphic functions and the integral, any biholomorphic map carries the metric of one domain to that of the other.\nFor the unit disc the Bergman metric is $\\frac{2 \\abs{dz}^2}{(1 - \\abs{z}^2)^2}$, a multiple of the hyperbolic metric.\n",
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