On the Ricci flow of homogeneous metrics on spheres

Annals of Global Analysis and Geometry - Tập 61 - Trang 499-517 - 2022
Sammy Sbiti1
1Department of Mathematics, University of Pennsylvania, Philadelphia, USA

Tóm tắt

We study the Ricci flow of the four-parameter family of Sp $$(n+1)$$ -invariant metrics on spheres. We determine their forward behaviour and also classify ancient solutions. In doing so, we exhibit a new one-parameter family of ancient solutions on spheres. This completes the classification of homogeneous ancient solutions on spheres.

Tài liệu tham khảo

Alekseevskii, D., Kinmel’fel’d, B.: Structure of homogeneous Riemannian manifolds with zero Ricci curvature. Funct. Anal. Appl. 9, 95–102 (1975) Bakas, I., Kong, S., Ni, L.: Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations. J. für die reine und angewandte Mathematik (Crelles J.) 2012(663), 209–248 (2012) Besse, A.L.: Einstein Manifolds. Springer, Berlin (2007) Böhm, C.: On the long time behavior of homogeneous Ricci flows. Commentarii Mathematici Helvetici 90(3), 543–571 (2015) Böhm, C., Wilking, B.: Manifolds with positive curvature operators are space forms. Ann. Math. 167(3), 1079–1097 (2008) Böhm, C., Wang, M., Ziller, W.: A variational approach for compact homogeneous n manifolds. Geom. Funct. Anal. 14(4), 681–733 (2004) Böhm, C., Lafuente, R., Simon, M.: Optimal curvature estimates for homogeneous Ricci flows. Int. Math. Res. Not. 2019(14), 4431–4468 (2019) Bourguignon, J.-P., Karcher, H.: Curvature operators: pinching estimates and geometric examples. In Annales scientifiques de l’École Normale Supérieure 11, 71–92 (1978) Brendle, S.: Ricci flow with surgery on manifolds with positive isotropic curvature. Ann. Math. 190(2), 465–559 (2019) Brendle, S., Kapouleas, N.: Gluing Eguchi–Hanson metrics and a question of page. Commun. Pure Appl. Math. 70(7), 1366–1401 (2017) Brendle, S., Schoen, R.: Manifolds with 1/4-pinched curvature are space forms. J. Am. Math. Soc. 22(1), 287–307 (2009) Brin, M., Stuck, G.: Introduction to Dynamical Systems. Cambridge University Press, Cambridge (2002) Buttsworth, T., Pulemotov, A., Rubinstein, Y. A., Ziller, W.: On the Ricci iteration for homogeneous metrics on spheres and projective spaces. Transform. Groups, 1–20 (2020) Buzano, M.: Ricci flow on homogeneous spaces with two isotropy summands. Ann. Glob. Anal. Geom. 45(1), 25–45 (2014) Cao, X., Saloff-Coste, L.: Backward Ricci flow on locally homogeneous 3-manifolds. Commun. Anal. Geom. 17(2), 305–325 (2009) Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow, vol. 77. American Mathematical Soc (2006) Enders, J., Mueller, R., Topping, P.: On type-I singularities in Ricci flow. Commun. Anal. Geom. 19(5), 905–922 (2011) Fateev, V.: The sigma model (dual) representation for a two-parameter family of integrable quantum field theories. Nucl. Phys. B 473(3), 509–538 (1996) Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982) Isenberg, J., Jackson, M.: Ricci flow of locally homogeneous geometries on closed manifolds. J. Differ. Geom. 35(3), 723–741 (1992) Jensen, G.R.: Einstein metrics on principal fibre bundles. J. Differ. Geom. 8(4), 599–614 (1973) Lafuente, R.A.: Scalar curvature behavior of homogeneous Ricci flows. J. Geom. Anal. 25(4), 2313–2322 (2015) Lauret, J.: Ricci flow of homogeneous manifolds. Math. Z. 274(1–2), 373–403 (2013) Lu, P., Wang, Y.: Ancient solutions of the Ricci flow on bundles. Adv. Math. 318, 411–456 (2017) Morgan, J.W., Tian, G.: Ricci Flow and the Poincaré Conjecture, vol. 3. American Mathematical Soc (2007) Naber, A.: Noncompact shrinking four solitons with nonnegative curvature. J. für die reine und angewandte Mathematik (Crelles J.) 2010(645), 125–153 (2010) Pediconi, F.: Diverging sequences of unit volume invariant metrics with bounded curvature. Ann. Glob. Anal. Geom. 56(3), 519–553 (2019) Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159 (2002) Petersen, P., Wylie, W.: On gradient Ricci solitons with symmetry. Proc. Am. Math. Soc. 137(6), 2085–2092 (2009) Verdiani, L., Ziller, W.: Positively curved homogeneous metrics on spheres. Math. Z. 261(3), 473–488 (2009) Wang, M.Y., Ziller, W.: Existence and non-existence of homogeneous Einstein metrics. Invent. Math. 84(1), 177–194 (1986) Ziller, W.: Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259(3), 351–358 (1982)