Abstract: We define regularity scales as alternative quantities of $\displaystyle \left(\max_{M} |Rm| \right)^{-1}$ to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson's conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded.

Abstract: Over the space of K\"ahler metrics associated to a fixed K\"ahler class, we first prove the lower bound of the energy functional $\tilde E^\beta$, then we provide the criteria of the geodesics rays to detect the lower bound of $\tilde {\mathfrak J}^\beta$-functional. They are used to obtain the $I$-properness of Mabuchi's $K$-energy. The criteria are examined by showing the convergence of the negative gradient flow of $\tilde {\mathfrak J}^\beta$-functional.

Abstract: We study the geodesic equation for the Dirichlet (gradient) metric in the space of K\"ahler potentials as a dynamical system. We first solve the initial value problem for the geodesic equation of the \emph{combination metric} including the gradient metric. We then discuss its application, such as energy identity, Jacobi fields and comparison theorem, curvature and the stability problem of the geodesic ray. As last, as the geometric motivation of the combination metric, we find that the Ebin metric restricted to the space of type II deformations of a Sasakian structure is the sum of the \emph{Calabi metric} and the gradient metric.

Abstract: In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics $\mathcal H_\beta$; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kähler metrics with cone singularities. Our approach concerns the generalization of the space defined by Donaldson to the case of Kähler manifolds with boundary; moreover we introduce a subspace $\mathcal H_C$ of $\mathcal H_\beta$ which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of $C^{1,1}_\beta$ geodesics whose boundary values lie in $\mathcal H_C$. Moreover, we prove that such geodesic is the limit of a sequence of $C^{2,\alpha}_\beta$ approximate geodesics under the $C^{1,1}_\beta$-norm. As a geometric application, we prove the metric space structure of $\mathcal H_C$.

Abstract: In this work we study the intrinsic geometry of the space of Kähler metrics under various Riemannian metrics and the corresponding variational structures. The first part is on the \emph{Dirichlet metric}. A motivation for the study of this metric comes from Chen-Zheng; there, they showed that the pseudo-Calabi flow is the gradient flow of the $K$-energy when $\mathcal{H}$ is endowed precisely with the Dirichlet metric. In this paper, we prove that the $K$-energy is convex at a constant scalar curvature Kähler (cscK) metric under the Dirichlet metric. That gives us the geometric explanation of the stability of the the pseudo-Calabi flow near a cscK metric in Chen-Zheng. The second part is on the family of weighted metrics, whose distinguished element is the Calabi metric studied in Calamai. We investigate as well their geometric properties. Then we focus on the constant weight metric. We use it to give an alternative proof of Calabi's uniqueness of the Kähler-Einstein metrics, when $C_1\leq0$. We also introduce a new functional called $G$-functional and we prove that its gradient flow has long time existence and also converges to the Kähler-Einstein metric when $C_1\leq 0$.

Abstract: The Kähler metric with cone singularities has been the main subject which is being studied recently. In this expository note, we focus on the modular space of the Kähler metric with cone singularities. We first summary our work on the construction of the geodesic of the cone singularities. Then we apply the cone geodesic to obtain a uniqueness theorem of the constant scalar curvature Kähler metrics with cone singularities.

Abstract: We first proved a compactness theorem of the Kähler metrics, which confirms a prediction in Chen. Then we prove several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed in Chen-Li-Wang to prove the small energy theorems of the Calabi flow.

Abstract: We first define the Pseudo-Calabi flow, as \begin{equation*} \left\{ \begin{aligned} {{\partial \varphi}\over {\partial t}}&= -f(\varphi), \\ \Delta_\varphi f(\varphi) &= S(\varphi) - \underline S. \\ \end{aligned} \right. \end{equation*} Then we prove the well-posedness of this flow including the short time existence, the regularity of the solution and the continuous dependence on the initial data. Next, we point out that the $L^\infty$ bound on Ricci curvature is an obstruction to the extension of the pseudo-Calabi flow. Finally, we show that if there is a cscK metric $\omega$ in its Kähler class, then for any initial potential in a small $C^{2,\alpha}$ neighborhood of this metric (defined in terms of the the $C^{2,\alpha}$ norm on the Kähler potential), the pseudo-Calabi flow must converge exponentially fast to a cscK metric near $\omega$ within the same Kähler class.

Abstract: We prove that on Kähler manifolds admit extremal metric $\omega$, for any Kähler potential $\varphi_0$ in its small $C^{2,\alpha}$ neighborhood, the Calabi flow will always stay in a small neighborhood of this extremal metric after pulling back by corresponding holomorphic transformations. Furthermore, when the initial data is $K_X$-invariant, the modified Calabi flow exponentially converge to a unique extremal metric nearby.

Abstract: In this paper, we prove that on a Fano manifold $M$ which admits a Kähler-Ricci soliton $(\omega,X)$, if the initial Kähler metric $\omega_{\varphi_0}$ is close to $\omega$ in some weak sense, then the weak Kähler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Particularly $\varphi_0$ is not assumed to be $K_X$-invariant. The methods based on the metric geometry of the space of the Kähler metrics are potentially applicable to other stability problem of geometric flows near the corresponding critical metrics.