arXiv research

The flow converges without Kähler-Einstein and develops ideal sheaves.

We introduce the inverse Monge-Ampere flow as the gradient flow of the Ding energy functional on the space of Kahler metrics in $2 πλc_1(X)$ for $λ=\pm 1$. We prove the long-time existence of the flow. In the canonically polarized case, we show that the flow converges smoothly to the unique Kahler-Einstein metric with …

S. K. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence the normalized Donaldson-Futaki invariants. We answer the question for the Ricci curvature formalism, in place of the scalar curvature. The principle is that the stability indicator is optimized by the multiplier ideal shea…

Study inverse boundary value problem for Monge-Ampère equation on convex domains.

The paper proves a comparison principle for complex Monge-Ampère flows and solves a uniqueness problem.

We present a deep generative model, named Monge-Ampère flow, which builds on continuous-time gradient flow arising from the Monge-Ampère equation in optimal transport theory. The generative map from the latent space to the data space follows a dynamical system, where a learnable potential function guides a compressible…

We study the regularizing properties of complex Monge-Ampère flows on a Kähler manifold $(X,ω)$ when the initial data are $ω$-psh functions with zero Lelong number at all points. We prove that the general Monge-Ampère flow has a solution which is immediately smooth. We also prove the uniqueness and stability of solutio…

Existence and uniqueness of bounded solutions to complex Monge-Ampère flows on Kähler manifolds.

Proves smooth solutions for generalised Monge-Ampère equations on projective manifolds.

Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampére equations. The purpose of this article, the second of a series on this subject, is to develop a viscosity theory for degenerate complex Mong…

The paper connects complex Monge-Ampère equations to $G_2$-structures on Calabi-Yau manifolds.

Solves complex Monge-Ampère equation for $(p,p)$-forms on Kähler manifolds.

The Ricci Calabi functional is a functional on the space of Kähler metrics of Fano manifolds. Its critical points are called generalized Kähler Einstein metrics. In this article, we show that the Hessian of the Ricci Calabi functional is non-negative at generalized Kähler Einstein metrics. As its application, we give a…

We study the parabolic flow for generalized complex Monge-Ampère type equations on closed Hermitian manifolds. We derive {\em a priori} $C^\infty$ estimates for normalized solutions, and then prove the $C^\infty$ convergence.

Study on Monge-Ampère equations with polynomial growth rates.

Proves solvability of general inverse σ_k equations with constant coefficients.

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

Enhances understanding of Kähler-Ricci flow singularities.

We prove that a general complex Monge-Ampère flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti-Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generaliza…

Study on singularities of Chern-Ricci flow on complex manifolds.

Proves long-term smoothness of curved surfaces evolving under specific curvature rules.

Derives $L^{\infty}$ estimate for Kähler-Ricci flows with weaker conditions.

We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.

Solves long-time solutions for a specific equation on hyperkähler manifolds.

The paper studies limits of flows on Kähler surfaces, proving convergence to solutions of equations.

Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.

In this article we study the Kähler Ricci flow, the corresponding parabolic Monge Ampère equation and complete non-compact Kähler Ricci flat manifolds. In our main result Theorem \ref{mainthm} we prove that if $(M, g)$ is sufficiently close to being Kähler Ricci flat in a suitable sense, then the Kähler Ricci flow \eqr…

In this paper, we study a class of fully nonlinear metric flow on Kähler 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 result of Song-Weinkove. As a consequence, under the given condition, we solved the…

Study solves complex equation on specific types of manifolds.

Study on stability and continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.

We prove uniform sup-norm estimates for the Monge-Ampere equation with respect to a family of Kahler metrics which degenerate towards a pull-back of a metric from a lower dimensional manifold. This is then used to show the existence of generalized Kahler-Einstein metrics as the limits of the Kahler-Ricci flow for some …

The paper studies a degenerate equation related to Kähler-Ricci flow on symplectic quotients.

Classifies surfaces translating under specific curvature flows.

We obtain higher order estimates for a parabolic flow on a compact Hermitian manifold. As an application, we prove that a bounded $\hatω$-plurisubharmonic solution of an elliptic complex Monge-Ampère equation is smooth under an assumption on the background Hermitian metric $\hatω$. This generalizes a result of Székelyh…

We study families of complex Monge-Ampère equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori $L^{\infty}$-estimates for the normalized solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has interesting consequences in the stud…

We study regularity properties of solutions to the Dirichlet problem for the complex Homogeneous Monge-Ampère equation. We show that for certain boundary data on $\mathbb P^1$ the solution $Φ$ to this Dirichlet problem is connected via a Legendre transform to an associated flow in the complex plane called the Hele-Shaw…

New criterion for solving inverse Hessian equations, including J-equation.

Proves existence and uniqueness of weak solutions for specific equations.

Defines and studies solutions to complex equations on Hermitian manifolds.

Study bi-harmonic flow with forcing term on smooth curves.

Estimates Kähler metric diameters with entropy bound alone.

We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

We establish a parabolic version of Tian's $C^{2,α}$-estimate for conical complex Monge-Ampere equations, which includes conical Kähler-Einstein metrics. Our estimate will complete the proof of the existence of unnormalized conical Kähler-Ricci flow in arXiv:1411.7284.

We study a parabolic complex Monge-Ampère type equation of the form \eqref{MA} on a complete noncompact \K manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain assumptions on a given real and closed (1,1) form $Ω$ and initial \K metric $g_0$ on…

We survey the Dirichlet problem for the complex Homogeneous Monge-Ampère Equation, both in the case of domains in $\mathbb C^n$ and the case of compact Kähler manifolds parametrized by a Riemann surface with boundary. We then give a self-contained account of previous work of the authors that connects this with the Hele…

Study inverse problems for twisted geodesic flows on manifolds.

Paper explores Monge-Ampère in deep learning and quantum geometry.

In this paper, we prove long time existence and convergence results for a class of general curvature flows with Neumann boundary condition. This is the first result for the Neumann boundary problem of non Monge-Ampere type curvature equations. Our method also works for the corresponding elliptic setting.