摘要
We first show that any 4-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy |Rm|<= cR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{{{\,\textrm{Rm}\,}}}|\le cR$$\end{document} for some positive constant c. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for 4-dimensional steady gradient solitons with linear scalar curvature decay and proper potential function. The technique is also used to establish a sufficient condition for a 3-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. As an application, we remove the positive curvature condition in a classification result by Math. Ann. 364(3-4), 777-792 (2014) in dimension three and show that any 3-dimensional gradient Ricci expander C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>2$$\end{document} asymptotic to C(S2),dt2+alpha t2gS2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( C(\mathbb S<^>2), dt<^>2+\alpha t<^>2 g_{\mathbb {S}<^>2}\right) $$\end{document} is rotationally symmetric, where alpha is an element of(0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1]$$\end{document} is a constant and gS2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mathbb {S}<^>2}$$\end{document} is the standard metric on S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}<^>2$$\end{document} with constant curvature 1.