Extending DiNezza-Lu's approach to the setting of big cohomology classes, we prove that solutions of degenerate complex Monge-Amp{è}re equations on compact K{ä}hler manifolds are continuous on a Zariski open set. This allows us to show that singular K{ä}hler-Einstein metrics on log canonical varieties of general type have continuous potentials on the ample locus outside of the non-klt part.
In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Ampère equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates for fully non-linear equations in Kähler geometry. As an application, we derive a uniform diameter bound for Kähler metrics satisfying certain Monge-Ampère equations.
In this note, we consider complex Monge-Ampere equation posed on a compact Kähler manifold. We show how to get $L^p$($p<\infty$) and $L^{\infty}$ estimate for the gradient of the solution in terms of the continuity of the right hand side.
Let $(X,ω)$ be a compact Kähler manifold. We prove that all Monge-Ampère capacities are comparable. Using this we give an alternative direct proof of the integration by parts formula for non-pluripolar products recently proved by M. Xia.
We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.
We consider degenerate Monge-Ampère equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampère operators. We then use the Perron method to solve Monge-Ampère equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoë, Caffarelli-Viaclovsky and Hultgren-Önnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow…
We prove a local volume noncollapsing estimate for Kähler metrics induced from a family of complex Monge-Ampère equations, assuming a local Ricci curvature lower bound. This local volume estimate can be applied to establish various diameter and gradient estimate.
We prove the bounded subsolution theorem for the complex Monge-Ampère type equation, with the right hand side being a positive Radon measure, on a compact Hermitian manifold with boundary.
A new proof for stability estimates for the complex Monge-Ampère and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations of the background metric in the case of the Monge-Ampère equation, and under degenerations to a big class in the case of Hessian equations.
In this paper, we consider the Monge-Ampère type equations on compact almost Hermitian manifolds. We derive $C^{\infty}$ a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. Finally, we obtain an existence result if there exists an admissible supersolution.
We discuss Monge-Ampère equations from the view point of differential geometry. It is known that a Monge-Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge-Ampère equations and prove that a $(k+1)$st order generalized Monge-Ampère equation corresponds to a special exterior differential system on a $k$-jet space. Then its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we verify that the Korteweg-de Vries (KdV) equation and the Cauchy-Riemann equations are …
These are the lecture notes for the Morningside Center of Mathematics Geometry Summer School on August 15-20, 2022. These lectures sketch the results by Yau, Demailly-Paun, the author, and Datar-Pingali about generalized Monge-Ampère equations and subvarieties and aim to use these results to study the Hodge conjecture.
We study the quaternionic Monge-Ampère equation on HKT manifolds admitting an HKT foliation having corank 4. We show that in this setting the quaternionic Monge-Ampère equation has always a unique solution for every basic datum. This approach includes the study of the equation on SU(3).
