Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions
Simon Brendle, Keaton Naff
Abstract
We extend the second part of \cite{Bre20} on the uniqueness of ancient $\kappa$-solutions to higher dimensions. In dimensions $n \geq 4$, an ancient $\kappa$-solution is a nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is $\kappa$-noncollapsed. We show that the only noncompact ancient $\kappa$-solutions up to isometry are a family of shrinking cylinders, a quotient thereof, or the Bryant soliton.
Topics & Concepts
MathematicsRicci flowUniquenessQuotientKappaBounded functionSymmetry (geometry)Pure mathematicsMathematical analysisFlow (mathematics)CurvatureRicci curvatureGeometryGeometric Analysis and Curvature FlowsGeometry and complex manifoldsAdvanced Differential Geometry Research