I heard in a conference that Yau's conjecture is open for positive Chern class. I read in an article that talked about some stability conditions necessary in this case. So I want to know if this stability condition is well-determined or still conjectural. More precisely, is there any precise statement of this conjecture or is it open ended?



Yau's conjecture for positive Chern class
A precise statement and proof of the relationship between stability and the existence of Calabi-Yau metrics is in:


1. arXiv:1302.0282, Xiuxiong Chen, Simon Donaldson, Song Sun, 
*Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof*
2. arXiv:1212.4714, Xiuxiong Chen, Simon Donaldson, Song Sun,
*Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π*
3. arXiv:1211.4566, Xiu-Xiong Chen, Simon Donaldson, Song Sun, 
*Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities*
4. arXiv:1210.7494, Xiu-Xiong Chen, Simon Donaldson, Song Sun
*Kahler-Einstein metrics and stability*
5. arXiv:1211.4669, Gang Tian, *K-stability and Kähler-Einstein metrics*


(so I understand; I haven't read all of this) 


