Time-dependent Sobolev Inequality along the Ricci Flow

d dt g = −2Rc g(0) = g0 on a closed manifold M. An important ingredient of Perelman’s proof of geometrization conjecture is the non-collapsing theorem of Ricci flow which makes sure that we can get a singularity model of the flow when a singularity exists. In [7], Zhang gave an easier way to prove the noncollapsing theorem of Ricci flow via a uniform Sobolev inequality along the flow. Unfortunately, there is a mistake in the proof of Zhang [7]. Later, Ye [6] corrected the error and Zhang [8] also corrected the error by himself. In [6] and [7], Ye and Zhang only considered Sobolev inequalities with L right hand side so that the surface case was excluded. In Hsu [2], she got uniform Sobolev inequalities with general right hand side so that the surface case was also included. In [6] and [2], Ye and Hsu got uniform Sobolev inequalities along the Ricci flow with the assumption that we are only considering Ricci flow in a finite time interval or that λ0(g0) > 0 where g0 is the initial metric of the Ricci flow and λ0(g) is the first eigenvalue of −∆g + R(g) 4 . Note that, we have the following evolution inequality of λ0 along the Ricci