Uniform Sobolev inequalities for manifolds evolving by Ricci flow

Let M be a compact n-dimensional manifold, $n\ge 2$, with metric g(t) evolving by the Ricci flow $\partial g_{ij}/\partial t=-2R_{ij}$ in (0,T) for some $T\in\Bbb{R}^+\cup\{\infty\}$ with $g(0)=g_0$. Let $\lambda_0(g_0)$ be the first eigenvalue of the operator $-\Delta_{g_0} +\frac{R(g_0)}{4}$ with respect to g_0. We extend a recent result of R. Ye and prove uniform logarithmic Sobolev inequality and uniform Sobolev inequalities along the Ricci flow for any $n\ge 2$ when either $T<\infty$ or $\lambda_0(g_0)>0$. As a consequence we extend Perelman's local $\kappa$-noncollapsing result along the Ricci flow for any $n\ge 2$ in terms of upper bound for the scalar curvature when either $T<\infty$ or $\lambda_0(g_0)>0$.