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Expanding Kähler-Ricci solitons coming out of Kähler cones. (English) Zbl 1510.53083

Summary: We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any \(n \in \mathbb{N}_0\) and for any negative line bundle \(L\) over a compact Kähler manifold \(D\), the total space of the vector bundle \(L^{\oplus (n+1)}\) admits a unique AC expanding gradient Kähler-Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if \(c_1 \bigg( K_D \oplus{(L^\ast)}^{\oplus (n+1)} \biggr) > 0\). This generalizes the examples already known in the literature. We further prove a general uniqueness result and show that the space of certain AC expanding gradient Kähler-Ricci solitons on \(\mathbb{C}^n\) with positive curvature operator on \((1, 1)\)-forms is path-connected.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53E30 Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows)
58D17 Manifolds of metrics (especially Riemannian)
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References:

[1] T. Aubin,Equations du type Monge-Amp‘´ere sur les vari´et´es k¨ahl´eriennes compactes, Bull. Sci. Math. (2)102(1978), no. 1, 63-95, MR0494932, Zbl 0374.53022. · Zbl 0374.53022
[2] C. Birkar, P. Cascini, C. Hacon, and J. McKernan,Existence of minimal models for varieties of log general type, J. Amer. Math. Soc.23(2010), no. 2, 405-468, MR 2601039, Zbl 1210.14019. · Zbl 1210.14019
[3] S. Boucksom, P. Eyssidieux, and V. Guedj,An introduction to the K¨ahler-Ricci flow, Lecture Notes in Mathematics, vol. 2086, Springer, Cham, 2013, pp. viii+333, MR3202578, Zbl 1276.53001. · Zbl 1276.53001
[4] C. P. Boyer and K. Galicki,Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008, MR2382957, Zbl 1155.53002. · Zbl 1155.53002
[5] R. L. Bryant,Gradient K¨ahler Ricci solitons, Ast´erisque (2008), no. 321, 51-97, G´eom´etrie diff´erentielle, physique math´ematique, math´ematiques et soci´et´e. I, MR2521644, Zbl 1183.53058.
[6] H.-D. Cao,Deformation of K¨ahler metrics to K¨ahler-Einstein metrics on compact K¨ahler manifolds, Invent. Math.81(1985), no. 2, 359-372, MR0799272, Zbl 0574.53042. · Zbl 0574.53042
[7] H.-D. Cao,Limits of solutions to the K¨ahler-Ricci flow, J. Differential Geom.45(1997), no. 2, 257-272, MR1449972, Zbl 0889.58067. · Zbl 0889.58067
[8] C.-W. Chen and A. Deruelle,Structure at infinity of expanding gradient Ricci soliton, Asian Journal of Mathematics19(2015), no. 5, 933-950, MR3431684, Zbl 1335.53083. · Zbl 1335.53083
[9] O. Chodosh and F. T.-H. Fong,Rotational symmetry of conical K¨ahler- Ricci solitons, Math. Ann.364(2016), nos 3-4, 777-792, MR3466851, Zbl 1336.53075. · Zbl 1336.53075
[10] R. J. Conlon and H.-J. Hein,Asymptotically conical Calabi-Yau manifolds, I, Duke Math. J.162(2013), no. 15, 2855-2902, MR3161306, Zbl 1283.53045. · Zbl 1283.53045
[11] B. Chow and D. Knopf,The Ricci flow: an introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004, MR2061425, Zbl 1200.53057. · Zbl 1086.53085
[12] B. Chow, P. Lu, and L. Ni,Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press, New York, 2006, MR2274812, Zbl 1118.53001. · Zbl 1118.53001
[13] T. C. Collins and G. Sz´ekelyhidi,The twisted K¨ahler-Ricci flow, J. Reine Angew. Math.716(2016), 179-205, MR3518375, Zbl 1357.53076. · Zbl 1357.53076
[14] T. C. Collins and V. Tosatti,A singular Demailly-P˘aun theorem, C. R. Math. Acad. Sci. Paris354(2016), no. 1, 91-95, MR3439731, Zbl 1344.32005. · Zbl 1344.32005
[15] S. Y. Cheng and S.-T. Yau,Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math.28 (1975), no. 3, 333-354, MR0385749, Zbl 0312.53031. · Zbl 0312.53031
[16] B.-L. Chen and X.-P. Zhu,Volume growth and curvature decay of positively curved K¨ahler manifolds, Q. J. Pure Appl. Math.1(2005), no. 1, 68-108, MR2154333, Zbl 1116.53045.
[17] A. Deruelle,Asymptotic estimates and compactness of expanding gradient Ricci solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)17(2017), no. 2, 485-530, MR3700376, Zbl 06761478. · Zbl 1379.53086
[18] A. Deruelle,Stability of non compact steady and expanding gradient Ricci solitons, Calc. Var. Partial Differential Equations54(2015), no. 2, 2367- 2405, MR3396456, Zbl 1325.53083. · Zbl 1325.53083
[19] A. Deruelle,Unique continuation at infinity for conical Ricci expanders, Int. Math. Res. Not. IMRN2017, no. 10, 3107-3147, MR3658133. · Zbl 1405.53070
[20] A. Deruelle,Smoothing out positively curved metric cones by Ricci expanders, Geom. Funct. Anal.26(2016), 188-249, MR3494489, Zbl 1343.53040. · Zbl 1343.53040
[21] A. S. Dancer and M. Y. Wang,On Ricci solitons of cohomogeneity one, Ann. Global Anal. Geom.39(2011), no. 3, 259-292, MR2769300, Zbl 1215.53040. · Zbl 1215.53040
[22] M. Feldman, T. Ilmanen, and D. Knopf,Rotationally symmetric shrinking and expanding gradient K¨ahler-Ricci solitons, J. Differential Geom. 65(2003), no. 2, 169-209, MR2058261, Zbl 1069.53036. · Zbl 1069.53036
[23] A. Futaki, H. Ono, and G. Wang,Transverse K¨ahler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Differential Geom.83 (2009), no. 3, 585-635, MR2581358, Zbl 1188.53042. · Zbl 1188.53042
[24] A. Futaki and M.-T. Wang,Constructing K¨ahler-Ricci solitons from Sasaki-Einstein manifolds, Asian J. Math.15(2011), no. 1, 33-52, MR2786464, Zbl 1222.53074. · Zbl 1222.53074
[25] R. Godement,Topologie alg´ebrique et th´eorie des faisceaux, Hermann, Paris, 1973, Troisi‘eme ´edition revue et corrig´ee, Publications de l’Institut de Math´ematique de l’Universit´e de Strasbourg, XIII, Actualit´es Scientifiques et Industrielles, No. 1252, MR0345092, Zbl 0080.16201. · Zbl 0275.55010
[26] R. S. Hamilton,The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.)7(1982), no. 1, 65-222, MR0656198, Zbl 0499.58003. · Zbl 0499.58003
[27] H.-J. Hein and C. LeBrun,Mass in K¨ahler Geometry, Comm. Math. Phys.347(2016), no. 1, 183-221, MR3543182, Zbl 1352.53060. · Zbl 1352.53060
[28] J. Koll´ar,Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007, MR2289519, Zbl 1113.14013. · Zbl 1113.14013
[29] A. Lunardi,Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients inRn, Studia Math.128(1998), no. 2, 171- 198, MR1490820, Zbl 0899.35014. · Zbl 0899.35014
[30] M. T. Lock and J. A. Viaclovsky,A sm¨org˚asbord of scalar-flat K¨ahler ALE surfaces, Journal f¨ur die reine und angewandte Mathematik (Crelles Journal), (2016). Retrieved 17 Oct. 2018, from doi:10.1515/crelle-20160007.
[31] J. Lott and P. Wilson,Note on asymptotically conical expanding Ricci solitons, Proc. Amer. Math. Soc.145(2017), no. 8, 3525-3529, MR3652804, Zbl 1366.53032. · Zbl 1366.53032
[32] D. H. Phong, N. Sesum, and J. Sturm,Multiplier ideal sheaves and the K¨ahler-Ricci flow, Comm. Anal. Geom.15(2007), no. 3, 613-632, MR2379807, Zbl 1143.53064. · Zbl 1143.53064
[33] M. Siepmann,Ricci flows of Ricci flat cones, Ph.D. thesis, ETH Z¨urich, 2013, available athttp://e-collection.library.ethz.ch/eserv/eth: 7556/eth-7556-02.pdf.
[34] G. Tian and X. Zhu,Uniqueness of K¨ahler-Ricci solitons, Acta Math. 184(2000), no. 2, 271-305, MR1768112, Zbl 1036.53052. · Zbl 1036.53052
[35] G. Tian and X. Zhu,A new holomorphic invariant and uniqueness of K¨ahler-Ricci solitons, Comment. Math. Helv.77(2002), no. 2, 297-325, MR1915043, Zbl 1036.53053. · Zbl 1036.53053
[36] G. Tian, S. Zhang, Z. Zhang, and X. Zhu,Perelman’s entropy and K¨ahler- Ricci flow on a Fano manifold, Trans. Amer. Math. Soc.365(2013), no. 12, 6669-6695, MR3105766, Zbl 1298.53037. · Zbl 1298.53037
[37] S.-T. Yau,On the Ricci curvature of a compact K¨ahler manifold and the complex Monge-Amp‘ere equation. I, Comm. Pure Appl. Math.31(1978), no. 3, 339-411, MR0480350, Zbl 0369.53059. · Zbl 0369.53059
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