Yau's theorem showed the existence. But I had difficulty finding examples other than complex tori. Any information will be appreciated.
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5$\begingroup$ You might be interested in A new proof of the existence of Kähler-Einstein metrics on K3 by P. Topiwala, Invent. Math. 89 (1987), no. 2, 425-454. It is somewhat more "explicit" than Yau's proof, though not as much as one could wish. $\endgroup$– abxSep 5, 2020 at 16:58
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4$\begingroup$ As far as I know, there is no explicit formula known for a Ricci-flat metric on a K3 surface. There have been some numerical studies done, and there are 'asymptotic' formulae for metrics near degenerations as well as some information for such metrics on K3 surfaces that have an elliptic fibration. See the papers of Gross and Wilson and later follow-up work on analogs of the SYZ conjecture in complex dimension 2. $\endgroup$– Robert BryantSep 5, 2020 at 17:12
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1$\begingroup$ I recall there being some work by Neitzke, Tripathy and others. $\endgroup$– user164740Sep 5, 2020 at 18:02
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$\begingroup$ Thanks for the helpful comments. How about the curvature tensor? Does there, by any chance, exist examples where the curvature tensor is explicit? $\endgroup$– clvolkovSep 6, 2020 at 0:36
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