The Mabuchi Geometry of Finite Energy Classes

By Tam'as Darvas. 2014

179 citations recorded by the source corpus.

We introduce different Finsler metrics on the space of smooth K\"ahler potentials that will induce a natural geometry on various finite energy classes $\mathcal E_{\tilde \chi}(X,\omega)$. Motivated by questions raised by R. Berman, V. Guedj and Y. Rubinstein, we characterize the underlying topology of these spaces in terms of convergence in energy and give applications of our results to existence of K\"ahler-Einstein metrics on Fano manifolds.

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

View Paper: https://www.semanticscholar.org/paper/261f21bfa424d43e6409ffd29a6608950ec3d23b

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Geometric pluripotential theory on Kähler manifolds (2019, background): “As detailed in this chapter, this conjecture was confirmed in [46], and later generalized to the case of L Finsler structures [47].”
Geodesic rays and stability in the cscK problem (2020, background): “Darvas [31] defined a Finsler-type d1-distance on E1(L) and proved that (E1(L), d1) is the metric completion of (H(L), d1) whose metric topology coincides with the strong topology introduced in [3].”
On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties (2017, background): “By Darvas’ work [16], E(1) can be characterized as the metric completion of H(χ) under a Finsler metric d1, where the Finsler metric d1 is defined as follows.”
Regularity of weak minimizers of the K-energy and applications to properness and K-stability (2016, methodology): “We refer to Sections 2.1-2.2 for more precise details about this metric structure introduced in [Da2].”
Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics (2014, methodology): “Building on [8] and the very recent work by Darvas and Guedj, [23], [24] and [32], we will show in [10] that the K-energy minimizing emanating from a given potential u0 in H(X,ω), gives rise to a curve of finite energy potentials in E (1)(X,ω) that we will call the finite energy Calabi flow with the”
On the constant scalar curvature Kähler metrics (II)—Existence results (2020, methodology): “Under an additional assumption that there exists a smooth cscK metric in the same Kähler class, Darvas-Rubinstein conjecture is verified in [5].”
On the constant scalar curvature Kähler metrics, general automorphism group (2018, methodology): “Darvas [24], one can complete H into a complete metric space (Ep, dp) and to this we can adjoin the sphere of infinity S̃∞ as in finite dimensional case.”