Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics

Abstract

In this article and in its sequel we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow, as well as another flow, in Kähler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics that arise as discretizations of these flows. We pose some problems regarding their dynamics. We point out a number of applications to well-studied objects in Kähler and conformal geometry such as constant scalar curvature metrics, Kähler-Ricci solitons, Nadel-type multiplier ideal sheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energy functionals and the geometry and structure of the space of Kähler metrics. E.g., we obtain a new sharp inequality strengthening the classical Moser-Trudinger-Onofri inequality on the two-sphere.

Multiplier ideal sheaves and integral invariants on toric Fano manifolds

“…Lemma 4.3 in [19]), there is a constant C such that (18) log(e hg det(u ij )) = −u + C 7 on U . Therefore…”

“…Multiplier ideal sheaves can also be constructed from the Kähler-Ricci flow ( [14], [17], [8]) and its discretization ( [16], [18]). As mentioned in Remark 1.4, we shall discuss the multiplier ideal sheaves constructed from the Kähler-Ricci flow on the toric del Pezzo surfaces.…”

The Moser-Trudinger-Onofri inequality

“…Also see [66] and in particular [66, Corollary 10.12] which contains a reinforced version of the inequality. In [24,Remark (1), p. 217], there is another proof which does not rely on symmetry, based on a result in [49].…”

Greatest lower bounds on Ricci curvature for Fano T-manifolds of complexity one

“…This invariant was first discussed, although not explicitly defined, by Tian in . It was first explicitly defined by Rubenstein in and was further studied by Szekelyhidi in . Tian's invariant is defined as follows: Definition Let be a Kähler manifold with .…”