The Mabuchi completion of the space of Kähler potentials

By Tam'as Darvas. 2014

107 citations recorded by the source corpus.

Suppose $(X,\omega)$ is a compact K\"ahler manifold. Following Mabuchi, the space of smooth K\"ahler potentials ${\cal H}$ can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space $({\cal H},d)$. We prove that the metric completion of $({\cal H},d)$ can be identified with $({\cal E}^2(X,\omega),{\tilde d})$, and this latter space is a complete non-positively curved geodesic metric space. In obtaining this result, we will rely on envelope techniques which allow for a treatment in a very general context. Profiting from this, we will characterize the pairs of potentials in ${\rm PSH}(X,\omega)$ that can be connected by weak geodesics and we will also give a characterization of ${\cal E}(X,\omega)$ in this context.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ee4b5a4e0ae6b545b8e6a55710645f1668a51bf9

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

On the constant scalar curvature Kähler metrics, general automorphism group (2018, background): “The main result of [10] and also the extension in [23] shows that (E2, d2) is non-positively curved.”
On the constant scalar curvature Kähler metrics (II)—Existence results (2020, methodology): “5 of [30]), Aubin’s J functional and the L1 distance are equivalent.”
On the singularity type of full mass currents in big cohomology classes (2016, result): “for. As a simple consequence of the above result, we can settle the conjecture of [BEGZ10, Remark 2.16] about the convexity of the classes Eχ(X,θ). A similar result in the Kähler case was obtained in [Dar14]. Corollary 2.15. Suppose χ∈ W−. Then E χ(X,θ) is convex. Proof. Given u0,u1 ∈ Eχ(X,θ) it fol”