Ancient Solutions of Ricci Flow with Type I Curvature Growth

Abstract

Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type I curvature growth. We show that a simply connected, noncompact, 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}-noncollapsed Type I ancient solution with nonnegative sectional curvature necessarily splits a Euclidean factor. It follows that a 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}-noncollapsed Type I ancient solution which is weakly PIC2 is a locally symmetric space.