Related papers: Heat Flows for Extremal K\"ahler Metrics

We prove that constant scalar curvature K\"ahler metric "adjacent" to a fixed K\"ahler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T…

We improve the description of $\mathbb{F}$-limits of noncollapsed Ricci flows in the K\"ahler setting. In particular, the singular strata $\mathcal{S}^k$ of such metric flows satisfy $\mathcal{S}^{2j}=\mathcal{S}^{2j+1}$. We also prove an…

In this paper, we mainly study the mean curvature flow in K\"ahler surfaces with positive holomorphic sectional curvatures. First, we prove that if the ratio $\lambda$ of the maximum and the minimum of the holomorphic sectional curvatures…

We investigate the emergence of finite-amplitude non-zonal flows on the sphere $\mathbb{S}^2$ arising from stationary solutions to the 2D Euler equations. By restricting the Laplace-Beltrami eigenspace to the invariant subspace of the…

In this paper, we use the modified potential well method to study the long time behaviors of solutions to the heat flow of H-system in a bounded smooth domain of $R^2$. Global existence and finite time blowup of solutions are proved when…

In this paper, we show that the Calabi flow can be extended as long as the $L^p$ scalar curvature is uniformly bounded for some $p>n$, and on a compact extremal K\"ahler manifold the Calabi flow with uniformly bounded $L^p(p>n)$ scalar…

Let $g(t)$, $t\in [0, +\infty)$, be a solution of the normalized K\"ahler-Ricci flow on a compact K\"ahler $n$-manifold $M$ with $c_{1}(M)>0$ and initial metric $g (0)\in 2\pi c_{1}(M)$. If there is a constant $C$ independent of $t$ such…

In this paper, we prove that on a Fano manifold $M$ which admits a K\"ahler-Ricci soliton $(\om,X)$, if the initial K\"ahler metric $\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak K\"ahler-Ricci flow exists globally and…

This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and…

An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis…

We study a class of fourth order curvature flows on a compact Riemannian manifold, which includes the gradient flows of a number of quadratic geometric functionals, as for instance the L2 norm of the curvature. Such flows can develop a…

We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature K\"ahler metrics to the case of extremal metrics. Namely, we show that a given…

In this paper, we introduce a new parabolic equation on K\"ahler manifolds. The static point of this flow is related to the existence of a lower bound of the Mabuchi energy. In this paper, we prove the flow always exists for all times for…

The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is ``no'' in the case of Euler and Navier-Stokes equations in dimension two. In dimension three…

We obtain a structure theorem for the group of holomorphic automorphisms of a conformally K\"ahler, Einstein-Maxwell metric, extending the classical results of Matsushima, Licherowicz and Calabi in the K\"ahler-Einstein, cscK, and extremal…

K\"ahler-Einstein currents, also known as singular K\"ahler-Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact K\"ahler spaces $X$ and their two defining…

We produce complete bounded curvature solutions to K\"ahler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth \K metric uniformly equivalent to another complete bounded curvature \K metric. We obtain…

In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat…

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…

We establish the well-posedness of the nonlocal mean curvature flow of order ${\alpha\in(0,1)}$ for periodic graphs on $\mathbb{R}^n$ in all subcritical little H\"older spaces ${\rm h}^{1+\beta}(\mathbb{T}^n)$ with $\beta\in(0,1)$.…