A gap theorem for minimal log discrepancies of non-canonical singularities in dimension three

We show that there exists a positive real number \δ>0 such that for any quasi-projective -Gorenstein $3$-fold X, if X has worse canonical singularities, that is, the minimal log discrepancy of X is than $1$, then the minimal log discrepancy of X is not greater than1-\δ. As applications, we show that the set of all non-canonical klt-Yau $3$-folds are bounded modulo flops, and the global indices of all Calabi-Yau $3$-folds are bounded from above.