Prescribing Ricci curvature on homogeneous spaces

References

[1] D. V. Alekseevskiĭ and B. N. Kimel’fel’d, Structure of homogeneous Riemannian spaces with zero Ricci curvature, Funkcional. Anal. i PriloŽen. 9 (1975), no. 2, 5–11; English translation: Funct. Anal. Appl. 9 (1975), 97–102. Search in Google Scholar

[2] R. Arroyo, M. Gould and A. Pulemotov, The prescribed Ricci curvature problem for naturally reductive metrics on non-compact simple Lie groups, preprint (2020), https://arxiv.org/abs/2006.15765. Search in Google Scholar

[3] R. Arroyo and R. Lafuente, On the signature of the Ricci curvature on nilmanifolds, preprint (2020), https://arxiv.org/abs/2009.11464. 10.1007/s00031-021-09686-5Search in Google Scholar

[4] R. Arroyo, A. Pulemotov and W. Ziller, The prescribed Ricci curvature problem for naturally reductive metrics on compact Lie groups, Differential Geom. Appl. 78 (2021), Paper No. 101794. 10.1016/j.difgeo.2021.101794Search in Google Scholar

[5] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer, Berlin 1987. 10.1007/978-3-540-74311-8Search in Google Scholar

[6] J. Bochnak, M. Coste, M.-F. Roy and M.-F. Roy, Real algebraic geometry, Ergeb. Math. Grenzgeb. (3) 36, Springer, Berlin 1998. 10.1007/978-3-662-03718-8Search in Google Scholar

[7] C. Böhm and R. Lafuente, Real geometric invariant theory, Differential geometry in the large. Part 1: Geometric evolution equations and curvature flow, Cambridge University Press, Cambridge (2020), 11–49. 10.1017/9781108884136.003Search in Google Scholar

[8] C. Böhm, M. Wang and W. Ziller, A variational approach for compact homogeneous Einstein manifolds, Geom. Funct. Anal. 14 (2004), no. 4, 681–733. 10.1007/s00039-004-0471-xSearch in Google Scholar

[9] G. E. Bredon, Introduction to compact transformation groups, Pure Appl. Math. 46, Academic Press, New York 1972. Search in Google Scholar

[10] T. Buttsworth, The prescribed Ricci curvature problem on three-dimensional unimodular Lie groups, Math. Nachr. 292 (2019), no. 4, 747–759. 10.1002/mana.201800052Search in Google Scholar

[11] T. Buttsworth and A. Pulemotov, The prescribed Ricci curvature problem for homogeneous metrics, Differential geometry in the large. Part 2: Structures on manifolds and mathematical physics, Cambridge University Press, Cambridge (2020), 169–192. 10.1017/9781108884136.010Search in Google Scholar

[12] T. Buttsworth, A. Pulemotov, Y. A. Rubinstein and W. Ziller, On the Ricci iteration for homogeneous metrics on spheres and projective spaces, Transform. Groups 26 (2021), no. 1, 145–164. 10.1007/s00031-020-09602-3Search in Google Scholar

[13] J. E. D’Atri and W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups, Mem. Amer. Math. Soc. 18 (1979), no. 215. 10.1090/memo/0215Search in Google Scholar

[14] E. Delay, Inversion d’opérateurs de courbures au voisinage d’une métrique Ricci parallèle II: variétés non compactes à géométrie bornée, Ark. Mat. 56 (2018), no. 2, 285–297. 10.4310/ARKIV.2018.v56.n2.a5Search in Google Scholar

[15] D. M. DeTurck, Metrics with prescribed Ricci curvature, Seminar on differential geometry, Ann. of Math. Stud. 102, Princeton University Press, Princeton (1982), 525–537. 10.1515/9781400881918-031Search in Google Scholar

[16] D. M. DeTurck, Prescribing positive Ricci curvature on compact manifolds, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), no. 3, 357–369. Search in Google Scholar

[17] D. M. DeTurck and N. Koiso, Uniqueness and nonexistence of metrics with prescribed Ricci curvature, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 5, 351–359. 10.1016/s0294-1449(16)30417-6Search in Google Scholar

[18] P. Eberlein and J. Heber, Quarter pinched homogeneous spaces of negative curvature, Internat. J. Math. 7 (1996), no. 4, 441–500. 10.1142/S0129167X96000268Search in Google Scholar

[19] C. S. Gordon, Naturally reductive homogeneous Riemannian manifolds, Canad. J. Math. 37 (1985), no. 3, 467–487. 10.4153/CJM-1985-028-2Search in Google Scholar

[20] M. Gould and A. Pulemotov, The prescribed Ricci curvature problem on homogeneous spaces with intermediate subgroups, preprint (2017), https://arxiv.org/abs/1710.03024. Search in Google Scholar

[21] R. Hamilton, The Ricci curvature equation, Seminar on nonlinear partial differential equations (Berkeley 1983), Math. Sci. Res. Inst. Publ. 2, Springer, New York (1984), 47–72. 10.1007/978-1-4612-1110-5_4Search in Google Scholar

[22] J. Heber, Noncompact homogeneous Einstein spaces, Invent. Math. 133 (1998), no. 2, 279–352. 10.1007/s002220050247Search in Google Scholar

[23] B. Kostant, On differential geometry and homogeneous spaces. I, II, Proc. Natl. Acad. Sci. USA 42 (1956), 258–261, 354–357. 10.1007/b94535_6Search in Google Scholar

[24] R. Lafuente and J. Lauret, Structure of homogeneous Ricci solitons and the Alekseevskii conjecture, J. Differential Geom. 98 (2014), no. 2, 315–347. 10.4310/jdg/1406552252Search in Google Scholar

[25] J. Lauret, Homogeneous nilmanifolds attached to representations of compact Lie groups, Manuscripta Math. 99 (1999), no. 3, 287–309. 10.1007/s002290050174Search in Google Scholar

[26] J. Lauret, Ricci flow of homogeneous manifolds, Math. Z. 274 (2013), no. 1–2, 373–403. 10.1007/s00209-012-1075-zSearch in Google Scholar

[27] J. Lauret, The search for solitons on homogeneous spaces, Geometry, Lie theory and applications. Abel Symposium 2019, Springer, New York (2022), in press. 10.1007/978-3-030-81296-6_8Search in Google Scholar

[28] J. Lauret and C. E. Will, On Ricci negative Lie groups, Geometry, Lie theory and applications. Abel Symposium 2019, Springer, New York (2022), in press. 10.1007/978-3-030-81296-6_9Search in Google Scholar

[29] J. Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293–329. 10.1016/S0001-8708(76)80002-3Search in Google Scholar

[30] Y. G. Nikonorov, The scalar curvature functional and homogeneous Einsteinian metrics on Lie groups, Sib. Math. J. 39 (1998), 504–509. 10.1007/BF02673906Search in Google Scholar

[31] A. Pulemotov, Metrics with prescribed Ricci curvature on homogeneous spaces, J. Geom. Phys. 106 (2016), 275–283. 10.1016/j.geomphys.2016.04.003Search in Google Scholar

[32] A. Pulemotov, Maxima of curvature functionals and the prescribed Ricci curvature problem on homogeneous spaces, J. Geom. Anal. 30 (2020), no. 1, 987–1010. 10.1007/s12220-019-00175-6Search in Google Scholar

[33] A. Pulemotov and Y. A. Rubinstein, Ricci iteration on homogeneous spaces, Trans. Amer. Math. Soc. 371 (2019), no. 9, 6257–6287.10.1090/tran/7498Search in Google Scholar

[34] R. Storm, The classification of 7- and 8-dimensional naturally reductive spaces, Canad. J. Math. 72 (2020), no. 5, 1246–1274. 10.4153/S0008414X19000300Search in Google Scholar

[35] C. Wang and M. Wang Instability of some Riemannian manifolds with real Killing spinors, Comm. Anal. Geom., to appear. Search in Google Scholar

[36] M. Y. Wang and W. Ziller, Existence and nonexistence of homogeneous Einstein metrics, Invent. Math. 84 (1986), no. 1, 177–194. 10.1007/BF01388738Search in Google Scholar