Related papers: Local solution and extension to the Calabi flow

We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally…

We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled…

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 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…

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…

We prove a uniform C^alpha estimate for collapsing Calabi-Yau metrics on the total space of a proper holomorphic submersion over the unit ball in C^m. The usual methods of Calabi, Evans-Krylov, and Caffarelli do not apply to this setting…

We study the local curvature estimates of long-time solutions to the normalized K\"ahler-Ricci flow on compact K\"ahler manifolds with semi-ample canonical line bundles. Using these estimates, we prove that on such a manifold, the set of…

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…

In this article, we study the higher-order regularity of the K\"ahler-Ricci flow on compact K\"ahler manifolds with semi-ample canonical line bundle. We proved, using a parabolic analogue of Hein-Tosatti's work on collapsing Calabi-Yau…

We consider a generalized Ricci flow with a given (not necessarily closed) three-form and establish the higher derivatives estimates for compact manifolds. As an application, we prove the compactness theorem for this generalized Ricci flow.…

We show the existence of a solution to the Ricci flow with a compact length space of bounded curvature, i.e., a space that has curvature bounded above and below in the sense of Alexandrov, as its initial condition. We show that this flow…

Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and $ \int_{\partial P} u ~ d \sigma < C_1, $ then there exists a constant…

In this paper we consider the Ricci flow on manifolds with boundary with appropriate control on its mean curvature and conformal class. We obtain higher order estimates for the curvature and second fundamental form near the boundary,…

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…

This paper presents a comprehensive study of the combinatorial $p$-th Calabi flow for both finite and infinite ideal circle patterns. In the finite case, we establish a sharp criterion: the combinatorial $p$-th Calabi flow with $p>1$…

We give a maximum principle proof of interior derivative estimates for the K\"ahler-Ricci flow, assuming local uniform bounds on the metric.

In this short note we prove that if the curvature tensor is uniformly bounded along the Calabi flow and the Mabuchi energy is proper, then the flow converges to a constant scalar curvature metric.

We consider the Cauchy problem for the nonlinear Schr\"odinger equation $iu_t+ \Delta u+ \lambda |u|^\alpha u=0$ in $\R^N $, in the $H^s$-subcritical and critical cases $0<\alpha \le 4/(N-2s)$, where $0<s<N/2$. Local existence of solutions…

We consider the formation of singularities along the Calabi flow with the assumption of the uniform Sobolev constant. In particular, on K\"ahler surface we show that any "maximal bubble" has to be a scalar flat ALE K\"ahler metric. In some…

In this paper we consider the local $L^p$ estimate of Riemannian curvature for the Ricci-harmonic flow or List's flow introduced by List \cite{List2005} on complete noncompact manifolds. As an application, under the assumption that the flow…