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Geodesic stability, the space of rays and uniform convexity in Mabuchi geometry

Tamás Darvas and Chinh H Lu

Geometry & Topology 24 (2020) 1907–1967
1 V Apostolov, D M J Calderbank, P Gauduchon, C W Tønnesen-Friedman, Hamiltonian 2–forms in Kähler geometry, III : Extremal metrics and stability, Invent. Math. 173 (2008) 547 MR2425136
2 T Aubin, Réduction du cas positif de l’équation de Monge–Ampère sur les variétés kählériennes compactes à la démonstration d’une inégalité, J. Funct. Anal. 57 (1984) 143 MR749521
3 K Ball, E A Carlen, E H Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math. 115 (1994) 463 MR1262940
4 R J Berman, B Berndtsson, Convexity of the K–energy on the space of Kähler metrics and uniqueness of extremal metrics, J. Amer. Math. Soc. 30 (2017) 1165 MR3671939
5 R J Berman, S Boucksom, P Eyssidieux, V Guedj, A Zeriahi, Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties, J. Reine Angew. Math. 751 (2019) 27 MR3956691
6 R J Berman, S Boucksom, V Guedj, A Zeriahi, A variational approach to complex Monge–Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 179 MR3090260
7 R Berman, S Boucksom, M Jonsson, A variational approach to the Yau–Tian–Donaldson conjecture, preprint (2015) arXiv:1509.04561
8 R J Berman, T Darvas, C H Lu, Convexity of the extended K–energy and the large time behavior of the weak Calabi flow, Geom. Topol. 21 (2017) 2945 MR3687111
9 R J Berman, T Darvas, C H Lu, Regularity of weak minimizers of the K–energy and applications to properness and K–stability, Ann. Sci. École Norm. Sup. 53 (2020) 267
10 B Berndtsson, Probability measures associated to geodesics in the space of Kähler metrics, from: "Algebraic and analytic microlocal analysis" (editors M Hitrik, D Tamarkin, B Tsygan, S Zelditch), Springer Proc. Math. Stat. 269, Springer (2018) 395 MR3903321
11 Z Błocki, The complex Monge–Ampère equation in Kähler geometry, from: "Pluripotential theory" (editors F Bracci, J E Fornæss), Lecture Notes in Math. 2075, Springer (2013) 95 MR3089069
12 Z Błocki, S Kołodziej, On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc. 135 (2007) 2089 MR2299485
13 S Boucksom, Variational and non-Archimedean aspects of the Yau–Tian–Donaldson conjecture, from: "Proceedings of the International Congress of Mathematicians, II" (editors B Sirakov, P N d Souza, M Viana), World Sci. (2018) 591 MR3966781
14 S Boucksom, D Eriksson, Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry, preprint (2018) arXiv:1805.01016
15 S Boucksom, P Eyssidieux, V Guedj, A Zeriahi, Monge–Ampère equations in big cohomology classes, Acta Math. 205 (2010) 199 MR2746347
16 S Boucksom, V Guedj, Regularizing properties of the Kähler–Ricci flow, from: "An introduction to the Kähler–Ricci flow" (editors S Boucksom, P Eyssidieux, V Guedj), Lecture Notes in Math. 2086, Springer (2013) 189 MR3202578
17 S Boucksom, T Hisamoto, M Jonsson, Uniform K–stability, Duistermaat–Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017) 743 MR3669511
18 S Boucksom, T Hisamoto, M Jonsson, Uniform K–stability and asymptotics of energy functionals in Kähler geometry, J. Eur. Math. Soc. 21 (2019) 2905 MR3985614
19 S Boucksom, M Jonsson, A non-Archimedean approach to K–stability, preprint (2018) arXiv:1805.11160
20 M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999) MR1744486
21 H Busemann, The geometry of geodesics, 6, Academic (1955) MR0075623
22 E Calabi, X X Chen, The space of Kähler metrics, II, J. Differential Geom. 61 (2002) 173 MR1969662
23 X Chen, On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Not. 2000 (2000) 607 MR1772078
24 X Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000) 189 MR1863016
25 X Chen, J Cheng, On the constant scalar curvature Kähler metrics : apriori estimates, preprint (2017) arXiv:1712.06697
26 X Chen, J Cheng, On the constant scalar curvature Kähler metrics : existence results, preprint (2018) arXiv:1801.00656
27 X Chen, J Cheng, On the constant scalar curvature Kähler metrics : general automorphism group, preprint (2018) arXiv:1801.05907
28 X Chen, T Darvas, W He, Compactness of Kähler metrics with bounds on Ricci curvature and functional, Calc. Var. Partial Differential Equations 58 (2019) MR3984099
29 X Chen, S Sun, Space of Kähler metrics, V : Kähler quantization, from: "Metric and differential geometry" (editors X Dai, X Rong), Progr. Math. 297, Birkhäuser (2012) 19 MR3220438
30 J Chu, V Tosatti, B Weinkove, On the C1,1 regularity of geodesics in the space of Kähler metrics, Ann. PDE 3 (2017) MR3695402
31 J A Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936) 396 MR1501880
32 T Darvas, The Mabuchi geometry of finite energy classes, Adv. Math. 285 (2015) 182 MR3406499
33 T Darvas, The Mabuchi completion of the space of Kähler potentials, Amer. J. Math. 139 (2017) 1275 MR3702499
34 T Darvas, Weak geodesic rays in the space of Kähler potentials and the class (X,ω), J. Inst. Math. Jussieu 16 (2017) 837 MR3680345
35 T Darvas, Geometric pluripotential theory on Kähler manifolds, from: "Advances in complex geometry" (editors Y A Rubinstein, B Shiffman), Contemp. Math. 735, Amer. Math. Soc. (2019) 1 MR3996485
36 T Darvas, E Di Nezza, C H Lu, L1 metric geometry of big cohomology classes, Ann. Inst. Fourier (Grenoble) 68 (2018) 3053 MR3959105
37 T Darvas, E Di Nezza, C H Lu, Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity, Anal. PDE 11 (2018) 2049 MR3812864
38 T Darvas, E Di Nezza, C H Lu, On the singularity type of full mass currents in big cohomology classes, Compos. Math. 154 (2018) 380 MR3738831
39 T Darvas, E Di Nezza, C H Lu, Log-concavity of volume and complex Monge–Ampère equations with prescribed singularity, Math. Ann. (2019)
40 T Darvas, E Di Nezza, H C Lu, The metric geometry of singularity types, J. Reine Angew. Math. (2020)
41 T Darvas, W He, Geodesic rays and Kähler–Ricci trajectories on Fano manifolds, Trans. Amer. Math. Soc. 369 (2017) 5069 MR3632560
42 T Darvas, L Lempert, Weak geodesics in the space of Kähler metrics, Math. Res. Lett. 19 (2012) 1127 MR3039835
43 T Darvas, C H Lu, Y A Rubinstein, Quantization in geometric pluripotential theory, Comm. Pure Appl. Math. 73 (2020) 1100
44 T Darvas, Y A Rubinstein, Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc. 30 (2017) 347 MR3600039
45 J P Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992) 361 MR1158622
46 J P Demailly, Regularization of closed positive currents of type (1,1) by the flow of a Chern connection, from: "Contributions to complex analysis and analytic geometry" (editors H Skoda, J M Trépreau), Aspects Math. E26, Vieweg (1994) 105 MR1319346
47 R Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. 2016 (2016) 4728 MR3564626
48 R Dervan, G Székelyhidi, The Kähler–Ricci flow and optimal degenerations, J. Differential Geom. 116 (2020) 187 MR4146359
49 E Di Nezza, V Guedj, Geometry and topology of the space of Kähler metrics on singular varieties, Compos. Math. 154 (2018) 1593 MR3830547
50 E Di Nezza, C H Lu, Uniqueness and short time regularity of the weak Kähler–Ricci flow, Adv. Math. 305 (2017) 953 MR3570152
51 S K Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, from: "Northern California Symplectic Geometry Seminar" (editors Y Eliashberg, D Fuchs, T Ratiu, A Weinstein), Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc. (1999) 13 MR1736211
52 S K Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002) 289 MR1988506
53 S K Donaldson, Scalar curvature and projective embeddings, II, Q. J. Math. 56 (2005) 345 MR2161248
54 A Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983) 437 MR718940
55 V Guedj, A Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005) 607 MR2203165
56 V Guedj, A Zeriahi, The weighted Monge–Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007) 442 MR2352488
57 V Guedj, A Zeriahi, Regularizing properties of the twisted Kähler–Ricci flow, J. Reine Angew. Math. 729 (2017) 275 MR3680377
58 W He, On the space of Kähler potentials, Comm. Pure Appl. Math. 68 (2015) 332 MR3298665
59 W He, On Calabi’s extremal metric and properness, Trans. Amer. Math. Soc. 372 (2019) 5595 MR4014289
60 J Jost, Nonpositive curvature: geometric and analytic aspects, Birkhäuser (1997) MR1451625
61 M Kell, Uniformly convex metric spaces, Anal. Geom. Metr. Spaces 2 (2014) 359 MR3290383
62 C O Kiselman, The partial Legendre transformation for plurisubharmonic functions, Invent. Math. 49 (1978) 137 MR511187
63 B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. 86 (1997) 115 MR1608566
64 S Kołodziej, The complex Monge–Ampère equation, Acta Math. 180 (1998) 69 MR1618325
65 K Kuwae, Jensen’s inequality on convex spaces, Calc. Var. Partial Differential Equations 49 (2014) 1359 MR3168636
66 C Li, C Xu, Special test configuration and K–stability of Fano varieties, Ann. of Math. 180 (2014) 197 MR3194814
67 T Mabuchi, Some symplectic geometry on compact Kähler manifolds, I, Osaka J. Math. 24 (1987) 227 MR909015
68 A Naor, L Silberman, Poincaré inequalities, embeddings, and wild groups, Compos. Math. 147 (2011) 1546 MR2834732
69 S i Ohta, Convexities of metric spaces, Geom. Dedicata 125 (2007) 225 MR2322550
70 D H Phong, J Sturm, The Monge–Ampère operator and geodesics in the space of Kähler potentials, Invent. Math. 166 (2006) 125 MR2242635
71 J Ross, D Witt Nyström, Analytic test configurations and geodesic rays, J. Symplectic Geom. 12 (2014) 125 MR3194078
72 S Semmes, Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114 (1992) 495 MR1165352
73 J Song, S Zelditch, Bergman metrics and geodesics in the space of Kähler metrics on toric varieties, Anal. PDE 3 (2010) 295 MR2672796
74 G Székelyhidi, Extremal metrics and K–stability, PhD thesis, University of London (2006) arXiv:math/0611002
75 G Tian, On Kähler–Einstein metrics on certain Kähler manifolds with C1(M) > 0, Invent. Math. 89 (1987) 225 MR894378
76 G Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990) 99 MR1064867
77 G Tian, The K–energy on hypersurfaces and stability, Comm. Anal. Geom. 2 (1994) 239 MR1312688
78 G Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1 MR1471884
79 G Tian, Canonical metrics in Kähler geometry, Birkhäuser (2000) MR1787650
80 D Witt Nyström, Monotonicity of non-pluripolar Monge–Ampère masses, Indiana Univ. Math. J. 68 (2019) 579 MR3951074
81 A Zeriahi, Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions, Indiana Univ. Math. J. 50 (2001) 671 MR1857051
82 K Zheng, Existence of constant scalar curvature Kähler cone metrics, properness and geodesic stability, preprint (2018) arXiv:1803.09506