Abstract
Using a stability criterion due to Kröncke, we show, providing \({n\ne 2k}\), the Kähler–Einstein metric on the Grassmannian \(Gr_{k}(\mathbb {C}^{n})\) of complex k-planes in an n-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kröncke and Knopf–Sesum on the instability of the Fubini–Study metric on \(\mathbb {CP}^{n}\) for \(n>1\). The key to the proof is using the description of Grassmannians as certain coadjoint orbits of SU(n). We are also able to prove that Kröncke’s method will not work on any of the other compact, irreducible, Hermitian symmetric spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berline, N., Vergne, M.: Fourier transforms of orbits of the coadjoint representation. Representation Theory of Reductive Groups (Park City, Utah: vol. 40 of Progress in Mathematics), pp. 53–67. Birkhäuser, Boston (1982)
Besse, A.L.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008). Reprint of the 1987 edition
Bott, R.: The geometry and representation theory of compact Lie groups. In: Luke, G.L., (ed.) Proceedings of the SRC/LMS Research Symposium held in Oxford, June 28–July 15, 1977 (1979), vol. 34 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, pp. v+341
Cao, H.D., Hamilton, R., Ilmanen, T.: Gaussian Densities and Stability for Some Ricci Solitons (2004). preprint, arXiv:math/0404165 [math.DG]
Cao, H.-D., He, C.: Linear stability of Perelman’s \(\nu \)-entropy on symmetric spaces of compact type. J. Reine Angew. Math. 709, 229–246 (2015)
Cao, H.-D., Zhu, M.: On second variation of Perelman’s Ricci shrinker entropy. Math. Ann. 353(3), 747–763 (2012)
Chow, B.: The Ricci flow on the \(2\)-sphere. J. Differ. Geom. 33(2), 325–334 (1991)
Coxeter, H.S.M.: The product of the generators of a finite group generated by reflections. Duke Math. J. 18, 765–782 (1951)
Duistermaat, J., Heckman, G.: On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69, 259–269 (1982)
Fulton, W., Harris, J.: Representation Theory. Graduate Texts in Mathematics, vol. 129. Springer, New York (1991). A first course, Readings in Mathematics
Gasqui, J., Goldschmidt, H.: Radon Transforms and the Rigidity of the Grassmannians. Annals of Mathematics Studies, vol. 156. Princeton University Press, Princeton (2004)
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Pure and Applied Mathematics. Wiley, New York (1978)
Hall, S.J.: The canonical Einstein metric on \({G}_{2}\) is dynamically unstable under the Ricci flow. Bull. Lond. Math. Soc. 51(3), 399–405 (2019)
Hall, S.J., Murphy, T.: On the linear stability of Kähler–Ricci solitons. Proc. Am. Math. Soc. 139(9), 3327–3337 (2011)
Hall, S.J., Murphy, T.: Variation of complex structures and the stability of Kähler–Ricci solitons. Pac. J. Math. 265(2), 441–454 (2013)
Hall, S.J., Murphy, T.: On the spectrum of the Page and the Chen–LeBrun–Weber metrics. Ann. Glob. Anal. Geom. 46(1), 87–101 (2014)
Hamilton, R.S.: The Ricci flow on surfaces. In: Aswi, A. (ed.) Mathematics and General Relativity. Contemporary Mathematics, vol. 1988, pp. 237–262. American Mathematics of Society, Providence (1986). (Santa Cruz, CA, : vol. 71 of
Haslhofer, R., Müller, R.: Dynamical stability and instability of Ricci-flat metrics. Math. Ann. 360(1–2), 547–553 (2014)
Humphreys, J.E.: Introduction to Lie algebras and Representation Theory. Graduate Texts in Mathematics, vol. 9. Springer, New York (1978). Second printing, revised
Isenberg, J., Knopf, D., Šešum, N.: Non-Kähler Ricci flow singularities modeled on Kähler–Ricci solitons. Pure Appl. Math. Q. 15(2), 749–784 (2019)
Kirillov, A.A.: Merits and demerits of the orbit method. Bull. Am. Math. Soc. 36(4), 433–488 (1999)
Knopf, D., Šešum, N.: Dynamic instability of \(\mathbb{CP}^{N}\) under Ricci flow. J. Geom. Anal. 29(1), 902–916 (2019)
Kröncke, K.: Stability and instability of Ricci solitons. Calc. Var. Partial Differ. Equ. 53(1–2), 265–287 (2015)
Kröncke, K.: Stability of Einstein metrics under Ricci flow. Commun. Anal. Geom. 28(2), 351–394 (2020)
Matsushima, Y.: Remarks on Kähler–Einstein manifolds. Nagoya Math. J. 46, 161–173 (1972)
Máximo, D.: On the blow-up of four-dimensional Ricci flow singularities. J. Reine Angew. Math. 692, 153–171 (2014)
McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Oxford Mathematical Monographs, 2nd edn. The Clarendon Press, New York (1998)
Paradan, P.-E.: The Fourier transform of semi-simple coadjoint orbits. J. Funct. Anal. 163(1), 152–179 (1999)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (2002). preprint, arXiv:math/0211159 [math.DG]
Rossmann, W.: Kirillov’s character formula for reductive Lie groups. Invent. Math. 48(3), 207–220 (1978)
Sesum, N.: Linear and dynamical stability of Ricci-flat metrics. Duke Math. J. 133(1), 1–26 (2006)
Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6, 274–304 (1954)
Tian, G., Zhu, X.: Convergence of the Kähler–Ricci flow on Fano manifolds. J. Reine Angew. Math. 678, 223–245 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hall, S.J., Murphy, T. & Waldron, J. Compact Hermitian Symmetric Spaces, Coadjoint Orbits, and the Dynamical Stability of the Ricci Flow. J Geom Anal 31, 6195–6218 (2021). https://doi.org/10.1007/s12220-020-00524-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-020-00524-w