Calabi conjecture | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Calabi_conjecture


00:01:58 1 Kähler&#8211;Einstein metrics
00:04:01 2 Outline of the proof of the Calabi conjecture
00:04:50 2.1 Transformation of the Calabi conjecture to a differential equation
00:07:41 2.2 Uniqueness of the solution
00:09:28 2.3 The set of iF/i is open
00:10:02 2.4 The set of iF/i is closed



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SUMMARY
=======
In mathematics, the Calabi conjecture was a conjecture about the existence of certain "nice" Riemannian metrics on certain complex manifolds, made by Eugenio Calabi (1954, 1957) and proved by Shing-Tung Yau (1977, 1978). Yau received the Fields Medal in 1982 in part for this proof.
The Calabi conjecture states that a compact Kähler manifold has a unique Kähler metric in the same class whose Ricci form is any given 2-form representing the first Chern class. In particular if the first Chern class vanishes there is a unique Kähler metric in the same class with vanishing Ricci curvature; these are called Calabi–Yau manifolds.
More formally, the Calabi conjecture states:

If M is a compact Kähler manifold with Kähler metric 
  
    
      
        g
      
    
    {\displaystyle g}
   and Kähler form 
  
    
      
        ω
      
    
    {\displaystyle \omega }
  , and R is any (1,1)-form representing the manifold's first Chern class, then there exists a unique Kähler metric 
  
    
      
        
          
            
              g
              ~
            
          
        
      
    
    {\displaystyle {\tilde {g}}}
   on M with Kähler form 
  
    
      
        
          
            
              ω
              ~
            
          
        
      
    
    {\displaystyle {\tilde {\omega }}}
   such that 
  
    
      
        ω
      
    
    {\displaystyle \omega }
   and 
  
    
      
        
          
            
              ω
              ~
            
          
        
      
    
    {\displaystyle {\tilde {\omega }}}
   represent the same class in cohomology H2(M,R) and the Ricci form of 
  
    
      
        
          
            
              ω
              ~
            
          
        
      
    
    {\displaystyle {\tilde {\omega }}}
   is R.The Calabi conjecture is closely related to the question of which Kähler manifolds have Kähler–Einstein metrics.