Related papers: On a class of fully nonlinear flow in K\"ahler geo…

By studying a complex Monge-Amp\`ere equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact K\"ahler manifold $N^n$ with $\Ric_k< 0$ for some integer $k$ with $1<k<n$, and the…

This paper is concerned with a fourth order nonlinear dispersive partial differential equation for closed curve flow on a K\"ahler manifold. The main results is that the initial value problem has a solution locally in time if the K\"ahler…

In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under…

We prove a sharp convergence theorem for the Yang-Mills flow on an $\mathrm{S}\mathrm{U}(r)$-bundle over a locally hyperK\"ahler ALE 4-manifold. Our main result is a noncompact version of the "parabolic gap theorem" previously established…

We study $n$-dimensional K\"ahler manifolds whose geodesic flows possess $n$ first integrals in involution that are fibrewise hermitian forms and simultaneously normalizable. Under some mild assumption, one can associate with such a…

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…

Uniform bounds are obtained using the auxiliary Monge-Amp\`ere equation method for solutions of very general classes of fully non-linear partial differential equations, assuming the existence of a ${C}$-subsolution in the sense of G.…

In this paper we survey the recent developments of the Ricci flows on complete noncompact K\"{a}hler manifolds and their applications in geometry.

We produce longtime solutions to the K\"ahler-Ricci flow for complete K\"ahler metrics on $\Bbb C ^n$ without assuming the initial metric has bounded curvature, thus extending results in [3]. We prove the existence of a longtime bounded…

$\,\,\,\,\,\,$In this paper, we prove that the nonautonomous Schr\"{o}dinger flow from a compact Riemannian manifold into a K\"ahler manifold admits a local solution. Under some certain conditions, the solution is unique and has higher…

In this paper, we set up a new Yamabe type flow on a compact Riemannian manifold $(M,g)$ of dimension $n\geq 3$. Let $\psi(x)$ be any smooth function on $M$. Let $p=\frac{n+2}{n-2}$ and $c_n=\frac{4(n-1)}{n-2}$. We study the Yamabe-type…

We consider the complex Monge-Amp\'{e}re equation on complete K\"{a}hler manifolds with cusp singularity along a divisor when the right hand side $F$ has rather weak regularity. We proved that when the right hand side $F$ is in some…

We show that the parabolic quaternionic Monge-Amp\`ere equation on a compact hyperk\"ahler manifold has always a long-time solution which once normalized converges smoothly to a solution of the quaternionic Monge-Amp\`ere equation. This is…

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 the present paper the exhaustive topological classification of nonsingular Morse-Smale flows of $n$-manifolds with two limit cycles is presented. Hyperbolicity of periodic orbits implies that among them one is attracting and another is…

In this paper, we construct local and global solutions to the K\"ahler-Ricci flow from a non-collapsed K\"ahler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the…

In this paper we study a normalised anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space R^n+1. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex…

Assuming Perelman's estimates, we give a new proof of uniform $L^\infty$ estimate along normalized K\"ahler-Ricci flow on Fano manifolds with K\"ahler-Einstein metrics, using Chen-Cheng's auxiliary Monge-Amp\`ere equation and the…

In this paper, we give an alternative proof for the convergence of K\"ahler-Ricci flow on a Fano mnaifold $(M,J)$. This proof differs from that in [TZ3]. Moreover, we generalize the main theorem of [TZ3] to the case that $(M,J)$ may not…

Sharp $L^\infty$ estimates are obtained for general classes of fully non-linear PDE's on non-K\"ahler manifolds, complementing the theory developed earlier by the authors in joint work with F. Tong for the K\"ahler case. The key idea is…