Uniqueness of compact ancient solutions to the higher-dimensional Ricci flow

In dimensions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>4</m:mn> </m:mrow> </m:math> {n≥ 4} , an ancient κ-solution is a nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is κ-noncollapsed. In this paper, we study the classification of ancient κ-solutions to n -dimensional Ricci flow on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {Sⁿ} , extending the result in [S. Brendle, P. Daskalopoulos and N. Sesum, Uniqueness of compact ancient solutions to three-dimensional Ricci flow, Invent. Math. 226 2021, 2, 579–651] to higher dimensions. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.