Related papers: Parabolic Flow for Generalized complex Monge-Amp\`…
Over many decades fully nonlinear PDEs, and the complex Monge-Amp\`ere equation in particular played a central role in the study of complex manifolds. Most previous works focused on problems that can be expressed through equations involving…
The application of equivalence method to classify Monge-Amp\`ere system leads to three orbits, parabolic case, hyperbolic case and elliptic case wich correspond to three types of Monge-Amp\`ere systems. In this paper we will study the…
We construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the…
We prove sharp uniform estimates for strong supersolutions of a large class of fully nonlinear degenerate elliptic complex equations. Our findings rely on ideas of Kuo and Trudinger who dealt with degenerate linear equations in the real…
In this paper, we generalize Medos-Wang's arguments and results on the mean curvature flow deformations of symplectomorphisms of $\CP^n$ in \cite{MeWa} to complex Grassmann manifold $G(n, n+m;\C)$ and compact totally geodesic…
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.…
We consider a general class of elliptic equations on hypercomplex manifolds which includes the quaternionic Monge-Amp\`ere equation, the quaternionic Hessian equation and the Monge-Amp\`ere equation for quaternionic $(n-1)$-plurisubharmonic…
We define a general class of elliptic equations for 2-forms on 4-manifolds, of which the complex Monge-Ampere equation is a prototype. We obtain some regularity results and discuss various connections (some speculative) with modern…
Studying the behavior of the K\"ahler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Amp\`ere equations. In this article, the third of a series on this subject,…
In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge-Amp\`ere equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic…
It is shown that the general solution of a homogeneous Monge-Amp\`{e}re equation in $n$-dimensional space is closely connected with the exactly (but only implicitly) integrable system \frac {\partial \xi_{j}}{\partial x_0}+\sum_{k=1}^{n-1}…
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…
We give a new proof of Brakke's partial regularity theorem up to C^{1,\varsigma} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The…
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…
In this paper, we study a class of fully nonlinear metric flow on K\"ahler manifolds, which includes the J-flow as a special case. We provide a sufficient and necessary condition for the long time convergence of the flow, generalizing the…
We adapt the PDE approach of Guo-Phong-Tong and Guo-Phong-Tong-Wang [17, 18] to prove an $L^\infty$ estimate for transverse complex Monge-Amp\`ere equations on homologically orientable transverse K\"ahler manifolds. As an application, we…
In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence…
We study the asymptotics of complete Kaehler-Einstein metrics on strictly pseudoconvex domains in C^n and derive a convergence theorem for solutions to the corresponding Monge-Ampere equation. If only a portion of the boundary is analytic,…
In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a…
We study the K\"ahler-Ricci flow on compact K\"ahler manifolds whose canonical bundle is big. We show that the normalized K\"ahler-Ricci flow has long time existence in the viscosity sense, is continuous in a Zariski open set, and converges…