Homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds

Böhm, Christoph¹ ; Lafuente, Ramiro A²

¹ Mathematisches Institut, University of Münster, Münster, Germany
² School of Mathematics and Physics, The University of Queensland, St Lucia campus, Brisbane, Australia

Geometry & topology, Tome 26 (2022) no. 2, pp. 899-936.

Résumé

We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using the fact that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit–Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.

Détail

DOI : 10.2140/gt.2022.26.899

Keywords: Einstein manifolds, homogeneous spaces, cohomogeneity-one manifold

Bibliographie

[1] D V Alekseevskii, Homogeneous Riemannian spaces of negative curvature, Mat. Sb. 96 (1975) 93

[2] D V Alekseevskii, B N Kimelfeld, Structure of homogeneous Riemannian spaces with zero Ricci curvature, Funkcional. Anal. i PriloŽen. 9 (1975) 5

[3] M T Anderson, A survey of Einstein metrics on 4–manifolds, from: "Handbook of geometric analysis, III" (editors L Ji, P Li, R Schoen, L Simon), Adv. Lect. Math. 14, International (2010) 1

[4] M T Anderson, P B Kronheimer, C Lebrun, Complete Ricci-flat Kähler manifolds of infinite topological type, Comm. Math. Phys. 125 (1989) 637 | DOI

[5] R M Arroyo, R Lafuente, Homogeneous Ricci solitons in low dimensions, Int. Math. Res. Not. 2015 (2015) 4901 | DOI

[6] R M Arroyo, R A Lafuente, The Alekseevskii conjecture in low dimensions, Math. Ann. 367 (2017) 283 | DOI

[7] T Aubin, Équations du type Monge–Ampère sur les variétés kähleriennes compactes, C. R. Acad. Sci. Paris Sér. A-B 283 (1976)

[8] L Bérard-Bergery, Sur la courbure des métriques riemanniennes invariantes des groupes de Lie et des espaces homogènes, Ann. Sci. École Norm. Sup. 11 (1978) 543 | DOI

[9] A L Besse, Einstein manifolds, 10, Springer (1987) | DOI

[10] O Biquard, Désingularisation de métriques d’Einstein, I, Invent. Math. 192 (2013) 197 | DOI

[11] O Biquard, Désingularisation de métriques d’Einstein, II, Invent. Math. 204 (2016) 473 | DOI

[12] C Böhm, Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces, Invent. Math. 134 (1998) 145 | DOI

[13] C Böhm, Non-compact cohomogeneity one Einstein manifolds, Bull. Soc. Math. France 127 (1999) 135

[14] C Böhm, Non-existence of cohomogeneity one Einstein metrics, Math. Ann. 314 (1999) 109 | DOI

[15] C Böhm, Non-existence of homogeneous Einstein metrics, Comment. Math. Helv. 80 (2005) 123 | DOI

[16] C Böhm, R A Lafuente, Immortal homogeneous Ricci flows, Invent. Math. 212 (2018) 461 | DOI

[17] C Böhm, R A Lafuente, Real geometric invariant theory, from: "Differential geometry in the large" (editors O Dearricott, W Tuschmann, Y Nikolayevsky, T Leistner, D Crowley), Lond. Math. Soc. Lect. Note Ser. 463, Cambridge Univ. Press (2020) 11 | DOI

[18] C Böhm, M Wang, W Ziller, A variational approach for compact homogeneous Einstein manifolds, Geom. Funct. Anal. 14 (2004) 681 | DOI

[19] C P Boyer, K Galicki, J Kollár, Einstein metrics on spheres, Ann. of Math. 162 (2005) 557 | DOI

[20] R L Bryant, Metrics with exceptional holonomy, Ann. of Math. 126 (1987) 525 | DOI

[21] R L Bryant, S M Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989) 829 | DOI

[22] X Chen, S Donaldson, S Sun, Kähler–Einstein metrics on Fano manifolds, I : Approximation of metrics with cone singularities, J. Amer. Math. Soc. 28 (2015) 183 | DOI

[23] X Chen, S Donaldson, S Sun, Kähler–Einstein metrics on Fano manifolds, II : Limits with cone angle less than 2π, J. Amer. Math. Soc. 28 (2015) 199 | DOI

[24] X Chen, S Donaldson, S Sun, Kähler–Einstein metrics on Fano manifolds, III : Limits as cone angle approaches 2π and completion of the main proof, J. Amer. Math. Soc. 28 (2015) 235 | DOI

[25] I Dotti Miatello, Transitive group actions and Ricci curvature properties, Michigan Math. J. 35 (1988) 427 | DOI

[26] J H Eschenburg, M Y Wang, The initial value problem for cohomogeneity one Einstein metrics, J. Geom. Anal. 10 (2000) 109 | DOI

[27] L Foscolo, M Haskins, New G2–holonomy cones and exotic nearly Kähler structures on S6 and S3 × S3, Ann. of Math. 185 (2017) 59 | DOI

[28] C S Gordon, M R Jablonski, Einstein solvmanifolds have maximal symmetry, J. Differential Geom. 111 (2019) 1 | DOI

[29] C S Gordon, E N Wilson, Isometry groups of Riemannian solvmanifolds, Trans. Amer. Math. Soc. 307 (1988) 245 | DOI

[30] K Grove, B Wilking, W Ziller, Positively curved cohomogeneity one manifolds and 3–Sasakian geometry, J. Differential Geom. 78 (2008) 33

[31] C He, P Petersen, W Wylie, Warped product Einstein metrics on homogeneous spaces and homogeneous Ricci solitons, J. Reine Angew. Math. 707 (2015) 217 | DOI

[32] J Heber, Noncompact homogeneous Einstein spaces, Invent. Math. 133 (1998) 279 | DOI

[33] S Helgason, Differential geometry, Lie groups, and symmetric spaces, 80, Academic (1978)

[34] J Hilgert, K H Neeb, Structure and geometry of Lie groups, Springer (2012) | DOI

[35] N Hitchin, Compact four-dimensional Einstein manifolds, J. Differential Geom. 9 (1974) 435

[36] M Jablonski, Homogeneous Ricci solitons are algebraic, Geom. Topol. 18 (2014) 2477 | DOI

[37] M Jablonski, Homogeneous Ricci solitons, J. Reine Angew. Math. 699 (2015) 159 | DOI

[38] M Jablonski, Strongly solvable spaces, Duke Math. J. 164 (2015) 361 | DOI

[39] M Jablonski, P Petersen, A step towards the Alekseevskii conjecture, Math. Ann. 368 (2017) 197 | DOI

[40] D D Joyce, Compact 8–manifolds with holonomy Spin(7), Invent. Math. 123 (1996) 507 | DOI

[41] D D Joyce, Compact Riemannian 7–manifolds with holonomy G2, I, J. Differential Geom. 43 (1996) 291 | DOI

[42] D D Joyce, Compact Riemannian 7–manifolds with holonomy G2, II, J. Differential Geom. 43 (1996) 329 | DOI

[43] D D Joyce, Compact manifolds with special holonomy, Oxford Univ. Press (2000)

[44] A W Knapp, Lie groups beyond an introduction, 140, Birkhäuser (2002)

[45] J L Koszul, Lectures on groups of transformations, 32, Tata Inst. Fund. Res. (1965)

[46] R A Lafuente, Scalar curvature behavior of homogeneous Ricci flows, J. Geom. Anal. 25 (2015) 2313 | DOI

[47] R Lafuente, J Lauret, Structure of homogeneous Ricci solitons and the Alekseevskii conjecture, J. Differential Geom. 98 (2014) 315

[48] J Lauret, Einstein solvmanifolds and nilsolitons, from: "New developments in Lie theory and geometry" (editors C S Gordon, J Tirao, J A Vargas, J A Wolf), Contemp. Math. 491, Amer. Math. Soc. (2009) 1 | DOI

[49] J Lauret, Einstein solvmanifolds are standard, Ann. of Math. 172 (2010) 1859 | DOI

[50] J Lauret, Ricci soliton solvmanifolds, J. Reine Angew. Math. 650 (2011) 1 | DOI

[51] J Lauret, C E Will, The Ricci pinching functional on solvmanifolds, Q. J. Math. 70 (2019) 1281 | DOI

[52] C Lebrun, Einstein metrics and Mostow rigidity, Math. Res. Lett. 2 (1995) 1 | DOI

[53] C Lebrun, Four-manifolds without Einstein metrics, Math. Res. Lett. 3 (1996) 133 | DOI

[54] J Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976) 293 | DOI

[55] A Naber, Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math. 645 (2010) 125 | DOI

[56] Y G Nikonorov, On the Ricci curvature of homogeneous metrics on noncompact homogeneous spaces, Sibirsk. Mat. Zh. 41 (2000) 421

[57] R S Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. 73 (1961) 295 | DOI

[58] P Petersen, W Wylie, On gradient Ricci solitons with symmetry, Proc. Amer. Math. Soc. 137 (2009) 2085 | DOI

[59] S Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992) 511 | DOI

[60] J A Thorpe, Some remarks on the Gauss–Bonnet integral, J. Math. Mech. 18 (1969) 779

[61] G Tian, K–stability and Kähler–Einstein metrics, Comm. Pure Appl. Math. 68 (2015) 1085 | DOI

[62] V S Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall (1974) | DOI

[63] M Y Wang, Einstein metrics from symmetry and bundle constructions, from: "Surveys in differential geometry : essays on Einstein manifolds" (editors C LeBrun, M Wang), Surv. Differ. Geom. 6, International (1999) 287 | DOI

[64] M Y K Wang, Einstein metrics from symmetry and bundle constructions: a sequel, from: "Differential geometry" (editors Y Shen, Z Shen, S T Yau), Adv. Lect. Math. 22, International (2012) 253

[65] M Y Wang, W Ziller, Existence and nonexistence of homogeneous Einstein metrics, Invent. Math. 84 (1986) 177 | DOI

[66] S T Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Comm. Pure Appl. Math. 31 (1978) 339 | DOI