## Abstract

A guiding principle in Kähler geometry is that the infinite-dimensional symmetric space Hℋ of Kähler metrics in a fixed Kähler class on a polarized projective Kähler manifold MM should be well approximated by finite-dimensional submanifolds Bk⊂Hℬk⊂ℋ of Bergman metrics of height kk (Yau, Tian, Donaldson). The Bergman metric spaces are symmetric spaces of type GC∕GGℂ∕G where G=U(dk+1)G=U(dk+1) for certain dkdk. This article establishes some basic estimates for Bergman approximations for geometric families of toric Kähler manifolds.

The approximation results are applied to the endpoint problem for geodesics of Hℋ, which are solutions of a homogeneous complex Monge–Ampère equation in A×XA×X, where A⊂CA⊂ℂ is an annulus. Donaldson, Arezzo and Tian, and Phong and Sturm raised the question whether Hℋ-geodesics with fixed endpoints can be approximated by geodesics of Bkℬk. Phong and Sturm proved weak C0C0-convergence of Bergman to Monge–Ampère geodesics on a general Kähler manifold. Our approximation results show that one has C2(A×X)C2(A×X) convergence in the case of toric Kähler metrics, extending our earlier result on CP1ℂℙ1.

The approximation results are applied to the endpoint problem for geodesics of Hℋ, which are solutions of a homogeneous complex Monge–Ampère equation in A×XA×X, where A⊂CA⊂ℂ is an annulus. Donaldson, Arezzo and Tian, and Phong and Sturm raised the question whether Hℋ-geodesics with fixed endpoints can be approximated by geodesics of Bkℬk. Phong and Sturm proved weak C0C0-convergence of Bergman to Monge–Ampère geodesics on a general Kähler manifold. Our approximation results show that one has C2(A×X)C2(A×X) convergence in the case of toric Kähler metrics, extending our earlier result on CP1ℂℙ1.

Original language | English (US) |
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Pages (from-to) | 295-358 |

Number of pages | 64 |

Journal | Analysis and PDE |

Volume | 3 |

Issue number | 3 |

State | Published - 2010 |