A simple proof on the non-existence of shrinking breathers for the Ricci   flow
  Suppose $M$ is a compact n-dimensional manifold, $n\ge 2$, with a metric $g_{ij}(x,t)$ that evolves by the Ricci flow $\partial_tg_{ij}=-2R_{ij}$ in $M\times (0,T)$. We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality. 
Shu-Yu Hsu