An important recent development in geometry has been the announcement of two claimed proofs of a long-standing conjecture about the existence of Kähler-Einstein metrics. Simon Donaldson is talking about this at MIT this week (see here and here), and the last in a series of his papers with Xiuxiong Chen and Song Sun giving details of their proof appeared on the arXiv earlier this week, see here. For the earlier papers in the series, see here and here, as well as the original announcement of the proof in outline here. Gang Tian also has a preprint with a proof, see here. As usual in mathematics, one might want to wait for these preprints to be refereed by experts before being sure that a proof is in hand.
Given any manifold, there’s an infinity of ways of putting a metric on it. A major theme in modern geometry and topology has been the pursuit of the idea that in many cases there may be a unique “best” choice for such a metric. The proof of the Poincaré Conjecture involved just this sort of idea, showing that starting with any metric on a simply-connected three-manifold one could deform it in a specific way to end up with certain special possibilities that could be completely analyzed.
For Kähler manifolds, the big open question of this kind has been that of whether one can find a unique metric that is both Kähler and Einstein (thus “Kähler-Einstein”). For negative first Chern class this was shown by Aubin and Yau, and for zero first Chern class by Yau in his proof of the Calabi conjecture (these are the “Calabi-Yau” manifolds). For positive Chern class there are counter-examples, but the conjecture has long been that Kähler manifolds satisfying an appropriate notion of “stability” will have such a unique Kähler-Einstein metric, and it is this conjecture that apparently has now been proven.
The details of this are far beyond my expertise, so I refer you to the papers quoted above, as well as some expository articles about the problem by Donaldson and Tian, as well as a series of blog posts (here, here, and here) by Terry Tao based on lectures by Yau.
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