Inspired by the canonical neighborhood theorem of G. Perelman in 3 dimensional, we study the weak compactness of sequence of ricci flow with scalar curvature bound, Kappa non-collapsing and integral curvature bound.

All of these constraints are natural in the Kahler ricci flow in Fano surface and as an application, we give a ricci flow based proof to the Calabi conjecture in Fano surface.

For an ergodic system, the theorem of Shannon-McMillan-Breiman states that for every finite generating partition the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere. In 1962 Ibragimov showed that the distribution of the measure of cylinder sets is lognormally distributed provided the measure is strong mixing and its conditional entropy function is sufficiently well approximable.
Carleson (1958) and Chung (1960) generalised the theorem of SMB to infinite partitions (provided the entropy is finite). We show that the measures of cylinder sets are lognormally distributed for uniformly strong mixing systems and infinite partitions and show that the rate of convergence is polynomial. Apart from the mixing property we require that a higher than fourth moment of the information function is finite. Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function. We also obtain the law of the iterated logarithm and the weak invariance principle for the information function.