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Dr. Jiawei Liu

Researcher


Otto-von-Guericke-Universität Magdeburg

E-mail: jiawei.liu(at)ovgu.de
Telephone: +49 391 67-51993

Publications within SPP2026

In this paper, we study the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle $2\pi\beta$ along the divisor, then for any $\beta'$ sufficiently close to $\beta$, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle $2\pi\beta'$ along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence before. As corollaries, we give parabolic proofs of Donaldson's openness theorem and his existence conjecture for the conical Kähler-Einstein metrics with positive Ricci curvatures.

 

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.

In this paper, we study the limiting flow of conical Kähler-Ricci flows when the cone angles tend to $0$. We prove the existence and uniqueness of this limiting flow with cusp singularity on compact Kähler manifold $M$ which carries a smooth hypersurface $D$ such that the twisted canonical bundle $K_M+D$ is ample. Furthermore, we prove that this limiting flow converge to a unique cusp Kähler-Einstein metric.

 

JournalAnnali di Matematica Pura ed Applicata (1923 -)
Link to preprint version

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.

In this paper, we study the long-time behavior of modified Calabi flow to study the existence of generalized Kähler-Ricci soliton. We first give a new expression of the modified $K$-energy and prove its convexity along weak geodesics. Then we extend this functional to some finite energy spaces. After that, we study the long-time behavior of modified Calabi flow.

 

JournalThe Journal of Geometric Analysis
Link to preprint version

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.

In this paper, by providing the uniform gradient estimates for approximating equations, we prove the existence, uniqueness and regularity of conical parabolic complex Monge-Ampère equation with weak initial data. As an application, we obtain a regularity estimate, that is, any $L^{\infty}$-solution of the conical complex Monge-Ampère equation admits the $C^{2,\alpha,\beta}$-regularity.

 

JournalCalculus of Variations and Partial Differential Equations
Link to preprint version

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.

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