Regularizing properties of the twisted Kähler-Ricci flow

Pluripotential kahler-ricci flows

We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log terminal singularities.

扭曲锥形凯勒-里奇流的正则化性质

Regularizing property of the twisted conical Kähler-Ricci flow

本文将光滑扭曲Kähler-Ricci流的正则化性质推广到锥形奇点情形，证明了从零Lelong数正闭电流出发的扭曲锥形Kähler-Ricci流具有正则性与唯一性。

Stability of solutions to complex Monge-Ampere flows

We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

Pluripotential Monge-Amp{è}re flows in Big Cohomology Classes

We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact K{ä}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential K{ä}hler-Ricci flows on compact K{ä}hler manifolds of general type as well as on stable varieties.

Regularizing properties of Complex Monge-Ampère flows II: Hermitian manifolds

We prove that a general complex Monge-Ampère flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti-Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted Chern-Ricci flow.