The Logarithmic Sobolev Inequality Along the Ricci Flow: The Case λ0(g0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _0(g_0)=0$$\end{document}

A uniform logarithmic Sobolev inequality, a uniform Sobolev inequality and a uniform κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-noncollapsing estimate along the Ricci flow are established in the situation that a certain smallest eigenvalue for the initial metric is zero.