相关论文: Degenerate J-flow on compact K\"ahler manifolds

We prove existence, uniqueness and convergence of solutions of the degenerate J-flow on Kahler surfaces. As an application, we establish the properness of the Mabuchi energy for Kahler classes in a certain subcone of the Kahler cone on…

The J-flow of S. K. Donaldson and X. X. Chen is a parabolic flow on Kahler manifolds with two Kahler metrics. It is the gradient flow of the J-functional which appears in Chen's formula for the Mabuchi energy. We find a positivity condition…

The J-flow is a parabolic flow on Kahler manifolds. It was defined by Donaldson in the setting of moment maps and by Chen as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. It is shown here that under…

Let $(X,\omega)$ be a compact connected K\"ahler manifold and denote by $(\mathcal E^p,d_p)$ the metric completion of the space of K\"ahler potentials $\mathcal H_\omega$ with respect to the $L^p$-type path length metric $d_p$. First, we…

We prove that if a pair of K\"ahler classes is $J$-nef, the $J$-flow on a compact K\"ahler surface converges to a weak solution of the Monge-Amp\`ere equation in the sense of currents. We also establish the same convergence behavior for the…

We study the J-flow on Kahler surfaces when the Kahler class lies on the boundary of the open cone for which global smooth convergence holds, and satisfies a nonnegativity condition. We obtain a C^0 estimate and show that the J-flow…

In this paper, we study a class of fully nonlinear metric flow on K\"ahler manifolds, which includes the J-flow as a special case. We provide a sufficient and necessary condition for the long time convergence of the flow, generalizing the…

In this paper, we study a family of twisted Calabi flows connecting the $J$-flow and Calabi flow on a compact K\"ahler manifold with a constant scalar curvature (cscK) metric. We show that for any initial data the twisted Calabi flow near…

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…

We develop a parabolic pluripotential theory on compact K{\"a}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{\`e}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor…

Donaldson defined a parabolic flow on Kahler manifolds which arises from considering the action of a group of symplectomorphisms on the space of smooth maps between manifolds. One can define a moment map for this action, and then consider…

Extending a recent theory developed on compact K\"ahler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Amp\`ere equations on compact Hermitian manifolds.…

We survey some recent developments on solutions of the K\"ahler-Ricci flow on compact K\"ahler manifolds which exist for all positive times.

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…

This paper investigates the twisted Calabi functional and the associated twisted Calabi flow on compact K\"ahler manifolds. Our main contributions are threefold: first, we establish the convexity of the twisted Calabi functional at its…

From the work of Dervan-Keller, there exists a quantization of the critical equation for the J-flow. This leads to the notion of J-balanced metrics. We prove that the existence of J-balanced metrics has a purely algebro-geometric…

We show that on a Kahler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kahler class. On toric manifolds we give a numerical characterization of when the J-flow converges, verifying a…

In this paper, we study the modified $J$-equation introduced by Li-Shi. We first show that, on compact K\"ahler manifolds, the solvability of the modified $J$-equation is equivalent to the coercivity of the modified $J$-functional.…

We show that for any solution to the K\"ahler-Ricci flow with positive bisectional curvature on a compact K\"ahler manifold $M^n$, the bisectional curvature has a uniform positive lower bound. As a consequence, the solution converges…

We introduced a new flow to the LYZ equation on a compact K\"ahler manifold. We first show the existence of the longtime solution of the flow. We then show that under the Collins-Jacob-Yau's condition on the subsolution, the longtime…