## Geometry & Topology

### Convexity of the extended K-energy and the large time behavior of the weak Calabi flow

#### Abstract

Let $(X,ω)$ be a compact connected Kähler manifold and denote by $(ℰp,dp)$ the metric completion of the space of Kähler potentials $ℋω$ with respect to the $Lp$–type path length metric $dp$. First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to $ℰp$ is a $dp$–lsc functional that is convex along finite-energy geodesics. Second, following the program of J Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space $(ℰ2,d2)$. This flow exists for all times and coincides with the usual smooth (twisted) Calabi flow whenever the latter exists. We show that the weak (twisted) Calabi flow either diverges with respect to the $d2$–metric or it $d1$–converges to some minimizer of the K-energy inside $ℰ2$. This gives the first concrete result about the long-time convergence of this flow on general Kähler manifolds, partially confirming a conjecture of Donaldson. We investigate the possibility of constructing destabilizing geodesic rays asymptotic to diverging weak (twisted) Calabi trajectories, and give a result in the case when the twisting form is Kähler. Finally, when a cscK metric exists in $ℋω$, our results imply that the weak Calabi flow $d1$–converges to such a metric.

#### Article information

Source
Geom. Topol., Volume 21, Number 5 (2017), 2945-2988.

Dates
Revised: 10 July 2016
Accepted: 22 October 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859278

Digital Object Identifier
doi:10.2140/gt.2017.21.2945

Mathematical Reviews number (MathSciNet)
MR3687111

Zentralblatt MATH identifier
1372.53073

#### Citation

Berman, Robert; Darvas, Tamás; Lu, Chinh. Convexity of the extended K-energy and the large time behavior of the weak Calabi flow. Geom. Topol. 21 (2017), no. 5, 2945--2988. doi:10.2140/gt.2017.21.2945. https://projecteuclid.org/euclid.gt/1510859278

#### References

• L Ambrosio, N Gigli, G Savaré, Gradient flows in metric spaces and in the space of probability measures, 2nd edition, Birkhäuser, Basel (2008)
• T Aubin, Équations du type Monge–Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. 102 (1978) 63–95
• M Bačák, The proximal point algorithm in metric spaces, Israel J. Math. 194 (2013) 689–701
• E Bedford, B A Taylor, The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math. 37 (1976) 1–44
• E Bedford, B A Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982) 1–40
• E Bedford, B A Taylor, Fine topology, Šilov boundary, and $(dd^c)^n$, J. Funct. Anal. 72 (1987) 225–251
• R J Berman, A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics, Adv. Math. 248 (2013) 1254–1297
• R J Berman, B Berndtsson, Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics, preprint (2014)
• R J Berman, S Boucksom, P Eyssidieux, V Guedj, A Zeriahi, Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties, preprint (2011)
• 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) 179–245
• R J Berman, T Darvas, C H Lu, Regularity of weak minimizers of the K-energy and applications to properness and K-stability, preprint (2016)
• R J Berman, H Guenancia, Kähler–Einstein metrics on stable varieties and log canonical pairs, Geom. Funct. Anal. 24 (2014) 1683–1730
• B Berndtsson, The openness conjecture and complex Brunn–Minkowski inequalities, from “Complex geometry and dynamics” (J E Fornæss, M Irgens, E F Wold, editors), Abel Symposia 10, Springer (2015) 29–44
• Z Błocki, On geodesics in the space of Kähler metrics, from “Advances in geometric analysis” (S Janeczko, J Li, D H Phong, editors), Adv. Lect. Math. 21, International Press, Somerville, MA (2012) 3–19
• Z Błocki, S Kołodziej, On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc. 135 (2007) 2089–2093
• T Bloom, N Levenberg, Pluripotential energy, Potential Anal. 36 (2012) 155–176
• S Boucksom, P Eyssidieux, V Guedj, A Zeriahi, Monge–Ampère equations in big cohomology classes, Acta Math. 205 (2010) 199–262
• M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer (1999)
• X Chen, On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices (2000) 607–623
• X Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000) 189–234
• X X Chen, A new parabolic flow in Kähler manifolds, Comm. Anal. Geom. 12 (2004) 837–852
• X Chen, Space of Kähler metrics, IV: On the lower bound of the K-energy, preprint (2008)
• X Chen, On the existence of constant scalar curvature Kähler metric: a new perspective, preprint (2015)
• X X Chen, W Y He, On the Calabi flow, Amer. J. Math. 130 (2008) 539–570
• X Chen, W He, The Calabi flow on toric Fano surfaces, Math. Res. Lett. 17 (2010) 231–241
• X Chen, M Paun, Y Zeng, On deformation of extremal metrics, preprint (2015)
• X Chen, S Sun, Calabi flow, geodesic rays, and uniqueness of constant scalar curvature Kähler metrics, Ann. of Math. 180 (2014) 407–454
• P T Chruściel, Semi-global existence and convergence of solutions of the Robinson–Trautman ($2$–dimensional Calabi) equation, Comm. Math. Phys. 137 (1991) 289–313
• T Darvas, The Mabuchi completion of the space of Kähler potentials, preprint (2015)
• T Darvas, The Mabuchi geometry of finite energy classes, Adv. Math. 285 (2015) 182–219
• T Darvas, W He, Geodesic rays and Kähler–Ricci trajectories on Fano manifolds, Trans. Amer. Math. Soc. (online publication March 2017)
• T Darvas, Y A Rubinstein, Tian's properness conjectures and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc. 30 (2017) 347–387
• E De Giorgi, New problems on minimizing movements, from “Boundary value problems for partial differential equations and applications” (J-L Lions, C Baiocchi, editors), Res. Notes Appl. Math. 29, Masson, Paris (1993) 81–98
• 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” (H Skoda, J-M Trépreau, editors), Aspects Math. E26, Vieweg, Braunschweig (1994) 105–126
• A Dembo, O Zeitouni, Large deviations: techniques and applications, 1st edition, Jones and Bartlett, Boston (1993)
• R Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. 2016 (2016) 4728–4783
• S Dinew, Uniqueness in $\mathscr{E}(X,\omega)$, J. Funct. Anal. 256 (2009) 2113–2122
• S K Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, from “Northern California Symplectic Geometry Seminar” (Y Eliashberg, D Fuchs, T Ratiu, A Weinstein, editors), Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc., Providence, RI (1999) 13–33
• S K Donaldson, Conjectures in Kähler geometry, from “Strings and geometry” (M Douglas, J Gauntlett, M Gross, editors), Clay Math. Proc. 3, Amer. Math. Soc., Providence, RI (2004) 71–78
• R Feng, H Huang, The global existence and convergence of the Calabi flow on $\mathbb{C}^n/\mathbb{Z}^n+i\mathbb{Z}^n$, J. Funct. Anal. 263 (2012) 1129–1146
• J Fine, Constant scalar curvature Kähler metrics on fibred complex surfaces, J. Differential Geom. 68 (2004) 397–432
• J Fine, Calabi flow and projective embeddings, J. Differential Geom. 84 (2010) 489–523
• Q Guan, X Zhou, A proof of Demailly's strong openness conjecture, Ann. of Math. 182 (2015) 605–616
• V Guedj, The metric completion of the Riemannian space of Kähler metrics, preprint (2014)
• V Guedj, A Zeriahi, The weighted Monge–Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007) 442–482
• V Guedj, A Zeriahi, Regularizing properties of the twisted Kähler–Ricci flow, preprint (2013)
• V Guedj, A Zeriahi, Degenerate complex Monge–Ampère equations, EMS Tracts in Math. 26, Eur. Math. Soc., Zürich (2017)
• W He, On the convergence of the Calabi flow, Proc. Amer. Math. Soc. 143 (2015) 1273–1281
• H Huang, Convergence of the Calabi flow on toric varieties and related Kähler manifolds, J. Geom. Anal. 25 (2015) 1080–1097
• H Huang, K Zheng, Stability of the Calabi flow near an extremal metric, Ann. Sc. Norm. Super. Pisa Cl. Sci. 11 (2012) 167–175
• W A Kirk, B Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008) 3689–3696
• S Kołodziej, The complex Monge–Ampère equation, Acta Math. 180 (1998) 69–117
• S Kołodziej, Hölder continuity of solutions to the complex Monge–Ampère equation with the right-hand side in $L^p$: the case of compact Kähler manifolds, Math. Ann. 342 (2008) 379–386
• H Li, B Wang, K Zheng, Regularity scales and convergence of the Calabi flow, preprint (2015)
• T Mabuchi, Some symplectic geometry on compact Kähler manifolds, I, Osaka J. Math. 24 (1987) 227–252
• U F Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom. 6 (1998) 199–253
• E D Nezza, C H Lu, Uniqueness and short time regularity of the weak Kähler–Ricci flow, preprint (2014)
• J Pook, Twisted Calabi flow on Riemann surfaces, Int. Math. Res. Not. 2016 (2016) 83–108
• S Semmes, Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114 (1992) 495–550
• J Stoppa, Twisted constant scalar curvature Kähler metrics and Kähler slope stability, J. Differential Geom. 83 (2009) 663–691
• J Streets, Long time existence of minimizing movement solutions of Calabi flow, Adv. Math. 259 (2014) 688–729
• J Streets, The consistency and convergence of $K$–energy minimizing movements, Trans. Amer. Math. Soc. 368 (2016) 5075–5091
• G Székelyhidi, Remark on the Calabi flow with bounded curvature, Univ. Iagel. Acta Math. 50 (2013) 107–115
• G Székelyhidi, V Tosatti, Regularity of weak solutions of a complex Monge–Ampère equation, Anal. PDE 4 (2011) 369–378
• V Tosatti, B Weinkove, The Calabi flow with small initial energy, Math. Res. Lett. 14 (2007) 1033–1039
• C Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58, Amer. Math. Soc., Providence, RI (2003)