Algebraic threefolds of general type with small volume

It is known that the optimal Noether inequality vol(X)>= 43pg(X)-103\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {vol} }(X) \ge \frac{4}{3}p_g(X) - \frac{10}{3}$$\end{document} holds for every 3-fold X of general type with pg(X)>= 11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_g(X) \ge 11$$\end{document}. In this paper, we give a complete classification of 3-folds X of general type with pg(X)>= 11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_g(X) \ge 11$$\end{document} satisfying the above equality by giving the explicit structure of a relative canonical model of X. This model coincides with the canonical model of X when pg(X)>= 23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_g(X) \ge 23$$\end{document}. We also establish the second and third optimal Noether inequalities for 3-folds X of general type with pg(X)>= 11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_g(X) \ge 11$$\end{document}. A novel phenomenon shows that there is a one-to-one correspondence between the three Noether inequalities and three possible residues of pg(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_g(X)$$\end{document} modulo 3. These results answer two open questions by Chen et al. (Duke Math J 169(9):1603-1164, 2020), and in dimension three an open question by Chen and Lai (Int J Math 31(1):2050005, 2020).