Abstract
Recently Darvas–Rubinstein proved a convergence result for the Kähler–Ricci iteration, which is a sequence of recursively defined complex Monge–Ampére equations. We introduce the Monge–Ampére iteration to be an analogous, but in a sense more general, sequence of recursively defined real Monge–Ampére second boundary value problems, and we establish sufficient conditions for its convergence. Each step in the iteration is a carefully chosen optimal transportation problem. We determine two cases where the convergence conditions are satisfied and provide geometric applications for both. First, we give a new proof of Darvas and Rubinstein’s general theorem on the convergence of the Ricci iteration in the case of toric Kähler manifolds, while at the same time generalizing their theorem to general convex bodies. Second, we introduce the affine iteration to be a sequence of prescribed affine normal problems and prove its convergence to an affine sphere. These give a new approach to recent existence and uniqueness results due to Berman–Berndtsson and Klartag.
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References
Alexandrov, A.D.: Dirichlet’s problem for the equation \(\det \,z_{ij} = \phi (z_1,\ldots,z_n,z,x_1,\ldots,x_n)\) I. Vestnik Leningrad Univ. Ser. Mat. Meh. Astr. 13(1), 5–24 (1958)
Berman, R.J., Berndtsson, B.: Real Monge–Ampère equations and Kähler–Ricci solitons on toric log Fano varieties. Ann. Fac. Sci. Toulouse Math. 22, 649–711 (2013)
Berman, R.J., Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties. J. Reine Angew. Math. (2016) (to appear)
Blaschke, W.: Vorlesungen über Differentialgeometrie II, Affine Differentialgeometrie. Springer, Berlin (1923)
Borell, C.: Convex set functions in d-space. Period. Math. Hungar. 6(2), 111–136 (1975)
Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)
Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44, 375–417 (1991)
Caffarelli, L.: Interior \({W}^{2, p}\) estimates for solutions of the Monge–Ampére equation. Ann. Math. 131, 135–150 (1990)
Caffarelli, L.: A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity. Ann. Math. 131, 129–134 (1990)
Caffarelli, L.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5, 99–104 (1992)
Calabi, E.: Complete affine hyperspheres I. Symp. Math. 10, 19–39 (1972)
Cheng, S.Y., Yau, S.T.: Complete affine hyperspheres part I: the completeness of affine metrics. Commun. Pure Appl. Math. 39, 839–866 (1986)
Cox, D., Little, J., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)
Darvas, T.: The Mabuchi geometry of finite energy classes. Adv. Math. 285, 182–219 (2015)
Darvas, T., Rubinstein, Y.: Convergence of the Kähler–Ricci iteration (2017). Preprint, arXiv:1705.06253
Deicke, A.: Uber die Finsler-räume mit \({A}_i=0\). Arch. Math. 4, 45–51 (1953)
Delzant, T.: Hamiltoniens périodiques et images convexes de lapplication moment. Bull. de la S.M.F 116, 315–339 (1988)
Donaldson, S.K.: Kähler geometry on toric manifolds and some other manifolds with large symmetry. Handbook of Geometric Analysis. Advanced Lectures in Mathematics, vol. 1, pp. 277–300. International Press of Boston, Somverville (2008)
Dubuc, S.: Critères de convexité et inégalités intégrales. Ann. Inst. Fourier (Grenoble) 27(1), 135–165 (1977)
Gigena, S.: Integral invariants of convex cones. J. Differ. Geom. 13, 191–222 (1978)
Guedj, V., Kolev, B., Yeganefar, N.: Kähler–Einstein fillings. J. Lond. Math. Soc. 88, 737–760 (2013)
Guillemin, V.: Kaehler structures on toric varieties. J. Differ. Geom. 40, 285–309 (1994)
Keller, J.: Ricci iterations on Kähler classes. J. Inst. Math. Jussieu 8, 743–768 (2009)
Klartag, B.: Affine hemispheres of elliptic type. Algebra i Analiz 29, 145–188 (2017)
Leindler, L.: On a certain converse of Hölders inequality. ii. Acta Sci. Math. (Szeged) 33, 217–223 (1972)
Lindsey, M., Rubinstein, Y.A.: Optimal transport via a Monge–Ampère optimization problem. SIAM J. Math. Anal. 49, 3073–3124 (2017)
Lutwak, E.: On some affine isoperimetric inequalities. J. Differ. Geom. 23, 1–13 (1986)
Lutwak, E.: Extended affine surface area. Adv. Math. 85, 39–68 (1991)
McCann, R.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80, 309–323 (1995)
Nomizu, K., Sasaki, T.: Affine Differential Geometry: Geometry of Affine Immersions. Cambridge University Press, Cambridge (1994)
Prékopa, A.: Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. (Szeged) 32, 301–315 (1971)
Prékopa, A.: On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34, 335–343 (1975)
Pulemotov, A., Rubinstein, Y.A.: Ricci iteration on homogeneous spaces (2016). Preprint, arXiv:1606.05064
Rauch, J., Taylor, B.A.: The Dirichlet problem for the multidimensional Monge–Ampère equation. Rocky Mt. J. Math. 7(2), 345–364 (1977)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rubinstein, Y.A.: The Ricci iteration and its applications. C. R. Acad. Sci. Paris 345, 445–448 (2007)
Rubinstein, Y.A.: Geometric quantization and dynamical constructions in the space of Kähler metrics. PhD thesis, Massachusetts Institute of Technology (2008)
Rubinstein, Y.A.: Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics. Adv. Math. 218, 1526–1565 (2008)
Rubinstein, Y.A.: Smooth and singular Kähler–Einstein metrics. Geometric and Spectral Analysis Contemporary Mathematics, vol. 630, pp. 45–138. AMS, Providence (2014)
Trudinger, N.S., Wang, X.-J.: The Monge–Ampére equation and its geometric applications. Handbook of Geometric Analysis, vol. 1, pp. 467–524. International Press, Vienna (2008)
Villani, C.: Optimal Transport Old and New. Springer, Berlin (2009)
Wang, X., Zhu, X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188, 87–103 (2004)
Wang, X.-J., Zhu, X.H.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188, 47–103 (2004)
Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation I. Commun. Pure Appl. Math. 31, 339–411 (1978)
Acknowledgements
Many thanks go to Y.A. Rubinstein for suggesting the Monge–Ampére iteration as a fruitful topic of study and for his indispensable guidance, inspiration, and encouragement; and to T. Darvas for his insights into the Kähler–Ricci iteration and for welcoming all my questions along the way. This research was also supported by the BSF Grant 2012236 and the NSF Grants DMS-1515703 and DMS-1440140.
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Hunter, R. Monge–Ampére iteration. Sel. Math. New Ser. 25, 73 (2019). https://doi.org/10.1007/s00029-019-0519-2
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DOI: https://doi.org/10.1007/s00029-019-0519-2
Keywords
- Real Monge–Ampére equation
- Kähler geometry
- Kähler–Einstein manifolds
- Ricci iteration
- Affine differential geometry
- Optimal transport