The Dirichlet and the weighted metrics for the space of Kähler metrics

Abstract

In this work we study the intrinsic geometry of the space of Kähler metrics under various Riemannian metrics and the corresponding variational structures. The first part is on the Dirichlet metric. A motivation for the study of this metric comes from Chen and Zheng (J Reine Angew Math, 674:195-251, 2013); there, Chen and the second author showed that the pseudo-Calabi flow is the gradient flow of the K -energy when H is endowed precisely with the Dirichlet metric. The second part is on the family of weighted metrics, whose distinguished element is the Calabi metric studied in Calamai (Math Ann 353:373-402, 2012). We investigate as well their geometric properties. Then we focus on the constant weight metric. We use it to give an alternative proof of Calabi's uniqueness of the Kähler-Einstein metrics (cf. Theorem 5.9), when C 1 ≤ 0. We also introduce a new functional called G-functional S.

Geometric pluripotential theory on Kähler manifolds

“…It is also possible to introduce a Dirichlet type Riemannian metric on H ω in terms of the gradient of the potentials [31,32]. Not much is known about the metric theory of this structure.…”

Comparison of the Calabi and Mabuchi geometries and applications to geometric flows

“…As we learned after the completion of this paper, in [14,Section 4], motivated by different goals, a family of Riemannian metrics was introduced and studied in detail that overlaps with our construction of L p,q -Calabi metrics when p = 2.…”

From the Kähler-Ricci flow to moving free boundaries and shocks

“…The metric 7.2 seems to first have appeared in [22], where it is attributed to Calabi and called Calabi's gradient metric (not to be confused with another metric usually referred to as the Calabi metric obtained by replacing the gradient of u by the Laplacian of u). The metric 7.2 was further studied in [26,27] where it is called the Dirichlet metric. See also [35] where the metric 7.2 appears from a symplecto-geometric point of view.…”