Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I

TL;DR

This paper explores discretizations of geometric evolution equations, particularly the Ricci flow in Kähler geometry, to study elliptic PDEs and dynamics on infinite-dimensional spaces, with applications to various geometric objects.

Contribution

It introduces and analyzes new discretizations of Ricci flow and related dynamics on the space of Kähler metrics, providing novel inequalities and insights.

Findings

New sharp inequality for the Moser-Trudinger-Onofri inequality

Discretizations yield insights into constant scalar curvature metrics

Applications to Kähler-Ricci solitons and energy functionals

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 Kahler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kahler 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 Kahler and conformal geometry such as constant scalar curvature metrics, Kahler-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 Kahler metrics. E.g., we obtain a new sharp inequality strengthening the classical Moser-Trudinger-Onofri inequality on the two-sphere.