  A question about Ricci flow is when the diameters of the manifold under the
evolving metrics stay finite and bounded away from 0.
  Topping \cite{T:1} addresses the question with an upper bound that depends on
the $L^{(n-1)/2}$ bound of the scalar curvature, volume and a local version of
Perelman's $\nu$ invariant. Here $n$ is the dimension. His result is sharp when
Perelman's F entropy is positive. In this note, we give a direct proof that for
all compact manifolds, the diameter bound depends just on the $L^{(n-1)/2}$
bound of the scalar curvature, volume and the Sobolev constants (or positive
Yamabe constant). This bound seems directly computable in large time for some
Ricci flows. In addition, since the result in its most general form is
independent of Ricci flow, further applications may be possible.
  A generally sharp lower bound for the diameters is also given, which depends
only on the initial metric, time and $L^\infty$ bound of the scalar curvature.
These results imply that, in finite time, the Ricci flow can neither turn the
diameter to infinity nor zero, unless the scalar curvature blows up.
