Related papers: Twisted Calabi functional and twisted Calabi flow

In this paper, we study the long-term behavior of the conical K\"ahler-Ricci flow on Fano manifold $M$. First, based on our work of locally uniform regularity for the twisted K\"ahler-Ricci flows, we obtain a long-time solution to the…

We study the convergence behavior of the general inverse $\sigma_k$-flow on K\"{a}hler manifolds with initial metrics satisfying the Calabi Ansatz. The limiting metrics can be either smooth or singular. In the latter case, interesting conic…

Using dynamical stability of symplectic curvature flow, we show that on a compact Calabi-Yau manifold, any small symplectic deformation of a K\"ahler form remains K\"ahler.

We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and…

Let us consider a projective manifold and $\Omega$ a volume form. We define the gradient flow associated to the problem of $\Omega$-balanced metrics in the quantum formalism, the \Omega$-balacing flow. At the limit of the quantization, we…

In this note, a modified K\"ahler-Ricci flow is introduced and studied. The main point is to show the flexibility of K\"ahler-Ricci flow and summarize some useful techniques.

We study the modified $J$-flow introduced in [15], particularly the singularities of the flow using the Calabi symmetry. In [20], on toric manifolds the convergence of modified $J$-flow to the smooth solution was proven under the assumption…

In this paper the dynamical stability of the Type IIA flow with no source near its stationary points is established. These stationary points had been shown previously by the authors to be Ricci-flat K\"ahler metrics on Calabi-Yau 3-folds.…

In this note, we study a degenerate twisted J-flow on compact K\"ahler manifolds. We show that it exists for all time, it is unique and converges to a weak solution of a degenerate twisted J-equation. In particular, this confirms an…

We consider the K\"ahler-Ricci flow on compact K\"ahler manifolds with semiample canonical bundle and intermediate Kodaira dimension, and show that the flow collapses to a canonical metric on the base of the Iitaka fibration in the locally…

We first proved a compactness theorem of the K\"ahler metrics, which confirms a prediction of Chen. Then we prove several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we…

We study the space of Sasaki metrics on a compact manifold $M$ by introducing an odd-dimensional analogue of the $J$-flow. That leads to the notion of critical metric in the Sasakian context. In analogy to the K\"ahler case, on a polarised…

In this paper, by limiting twisted conical K\"ahler-Ricci flows, we prove the long-time existence and uniqueness of cusp K\"ahler-Ricci flow on compact K\"ahler manifold $M$ which carries a smooth hypersurface $D$ such that the twisted…

We investigate how to obtain various flows of K\"ahler metrics on a fixed manifold as variations of K\"ahler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that…

We study the behaviour of the normalized K\"ahler-Ricci flow on complete K\"ahler manifolds of negative holomorphic sectional curvature. We show that the flow exists for all time and converges to a K\"ahler-Einstein metric of negative…

We show that on Kahler manifolds M with c_1(M)=0 the Calabi flow converges to a constant scalar curvature metric if the initial Calabi energy is sufficiently small. We prove a similar result on manifolds with c_1(M)<0 if the Kahler class is…

In this paper, we prove that there exists a dimensional constant $\delta > 0$ such that given any background K\"ahler metric $\omega$, the Calabi flow with initial data $u_0$ satisfying \begin{equation*} \partial \bar \partial u_0 \in…

Suppose $(X,\omega)$ is a compact K\"ahler manifold. We introduce and explore the metric geometry of the $L^{p,q}$-Calabi Finsler structure on the space of K\"ahler metrics $\mathcal H$. After noticing that the $L^{p,q}$-Calabi and…

We prove the longtime existence and convergence of the Calabi flow on toric Fano surfaces in a large family of Kahler classes where the class has positive extremal Hamiltonian potential and the initial Calabi energy is bounded by some…

We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X. We construct a natural barrier function along the flow, and introduce some techniques to study the blow-up of the curvature along the flow.…