相关论文: The Calabi flow on K\"ahler surface with bounded S…

Assuming uniform bounds for the curvature, the exponential convergence of the K\"ahler-Ricci flow is established under two conditions which are a form of stability: the Mabuchi energy is bounded from below, and the dimension of the space of…

We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of…

If a normalized K\"{a}hler-Ricci flow $g(t),t\in[0,\infty),$ on a compact K\"{a}hler $n$-manifold, $n\geq 3$, of positive first Chern class satisfies $g(t)\in 2\pi c_{1}(M)$ and has $L^{n}$ curvature operator uniformly bounded, then the…

Motivated by the problem of finding constant scalar curvature K\"ahler metrics, we investigate a Ricci iteration sequence of Rubinstein that discretizes the pseudo-Calabi flow. While the long time existence of the flow is still an open…

Let $D$ be a smooth divisor in a compact complex manifold $X$ and let $\beta \in (0,1)$. We show that in any positive co-homology class on $X$ there is a K\"ahler metric with cone angle $2\pi\beta$ along $D$ which has bounded Ricci…

A compactness theorem is proved for a family of K\"{a}hler surfaces with constant scalar curvature and volume bounded from below, diameter bounded from above, Ricci curvature bounded and the signature bounded from below. Furthermore, a…

The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal…

We study some estimates along the Kahler Ricci flow on Fano manifolds. Using these estimates, we show the convergence of Kahler Ricci flow directly if the $\alpha$-invariant of the canonical class is greater than $\frac{n}{n+1}$. Applying…

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 question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent…

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 show that the twisted K\"ahler-Ricci flow on a complex manifold X converges to a flow of moving free boundaries, in a certain scaling limit. This leads to a new phenomenon of singularity formation and topology change which can be seen as…

We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in…

Inspired by recent work of S. K. Donaldson on constant scalar curvature metrics on toric complex surfaces, we study obstructions to the extension of the Calabi flow on a polarized toric variety. Under some technical assumptions, we prove…

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…

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature K\"ahler metric. This…

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the…

We establish a uniform Sobolev inequality for K\"ahler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular K\"ahler metrics on K\"ahler spaces with…

An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set…

In this article, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal K\"ahler metric. In fact, this extremal K\"ahler metric can even be taken…