Weak solutions to degenerate complex Monge-Amp\'ere Flows II
Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi
TL;DR
This paper develops a viscosity theory for degenerate complex Monge-Ampère flows on compact Kähler manifolds, enabling the study of the normalized Kähler-Ricci flow on varieties with canonical singularities.
Contribution
It introduces a new viscosity framework for degenerate Monge-Ampère flows, extending the analysis of Kähler-Ricci flow to mildly singular varieties.
Findings
Established a viscosity theory for degenerate complex Monge-Ampère flows.
Generalized the normalized Kähler-Ricci flow to varieties with canonical singularities.
Provided tools for long-term behavior analysis of these flows.
Abstract
Studying the (long-term) behavior of the K\"ahler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Amp\'ere equations. The purpose of this article, the second of a series on this subject, is to develop a viscosity theory for degenerate complex Monge-Amp\'ere flows on compact K\"ahler manifolds. Our general theory allows in particular to define and study the (normalized) K\"ahler-Ricci flow on varieties with canonical singularities, generalizing results of Song and Tian.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Algebraic Geometry and Number Theory