  We develop some estimates under the Ricci flow and use these estimates to
study the blowup rates of curvatures at singularities. As applications, we
obtain some gap theorems:
  $\displaystyle \sup_X |Ric|$ and $\displaystyle \sqrt{\sup_X |Rm|} \cdot
\sqrt{\sup_X |R|}$
  must blowup at least at the rate of type-I. Our estimates also imply some gap
theorems for shrinking Ricci solitons.
