With respect to curvature and hyperbolicity it should be noted however that while curvature is a property that is essentially local, hyperbolicity is a large-scale property which does not see local ( i . e . happening in a bounded region ) metric phenomena.
22.
This appeared in the research article Global Hyperbolicity of Renormalization for Smooth Unimodal Mappings published at the journal Annals of Mathematics ( 2006 ) and was based in particular in the previous works of Sandy Davie, Dennis Sullivan, Curtis T . McMullen and Mikhail Lyubich.
23.
In view of the initial value formulation for Einstein's equations, global hyperbolicity is seen to be a very natural condition in the context of general relativity, in the sense that given arbitrary initial data, there is a unique maximal globally hyperbolic solution of Einstein's equations.
24.
For example, he gave a new proof of a conjecture of Andr?Bloch ( 1926 ) about holomorphic curves in closed subvarieties of Abelian varieties, proved a conjecture of Shoshichi Kobayashi ( about the Kobayashi-hyperbolicity of generic hypersurfaces of high degree in projective " n "-dimensional space ) in the three-dimensional case and achieved partial results on a conjecture of Mark Green and Phillip Griffiths ( which states that a holomorphic curve on an algebraic surface of general type with c _ { 1 } ^ 2 > c _ 2 cannot be Zariski-dense ).
