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

By Yanir A. Rubinstein. 2007

95 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/92dfffb7f95ce5ba240ead7486bd781b43235e41

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In Applied Mathematics

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Geometric pluripotential theory on Kähler manifolds (2019, methodology): “In the next lemma, we will make us of the inverse Ricci operator, introduced in [92] in connection with the so-called Ricci iteration.”
Ricci flow on surfaces with conic singularities (2013, methodology): “ simpler and more conceptual. Finally, as already noted, the Ka¨hler–Ricci ﬂow motivated a new Ricci continuity method and its discrete analogue, the Ricci iteration, both of which were introduced in [36]. These were employed in [27, 20, 26] to study Ka¨hler–Einstein edge metrics. This paper has sev”
A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics (2010, methodology): “As for the functional G(u) it generalizes a functional introduced by Ding [39] in Kähler-Einstein geometry that we following [73] will refer to as the Ding functional.”
Kahler-Einstein metrics with edge singularities (2011, methodology): “In particular, we employ a particular continuity method closely related to the Ricci iteration [42], that together with the Chern–Lu inequality allows us to obtain the a priori estimate on the Laplacian assuming only that the reference edge metric has bisectional curvature bounded above.”