Complex Monge-Ampere Equation Research Articles

Octonionic Calabi–Yau Theorem

On a certain class of 16-dimensional manifolds a new class of Riemannian metrics, called octonionic Kähler, is introduced and studied. It is an octonionic analogue of Kähler metrics on complex manifolds and of HKT-metrics of hypercomplex manifolds. Then for this class of metrics an octonionic version of the Monge–Ampère equation is introduced and solved under appropriate assumptions. The latter result is an octonionic version of the Calabi–Yau theorem from Kähler geometry.

Octonionic Calabi–Yau Theorem

On a certain class of 16-dimensional manifolds a new class of Riemannian metrics, called octonionic Kähler, is introduced and studied. It is an octonionic analogue of Kähler metrics on complex manifolds and of HKT-metrics of hypercomplex manifolds. Then for this class of metrics an octonionic version of the Monge–Ampère equation is introduced and solved under appropriate assumptions. The latter result is an octonionic version of the Calabi–Yau theorem from Kähler geometry.

Moser-Trudinger inequalities and complex Monge-Ampère equation

Our aim is to give a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, our result already gives a new Moser-Trudinger inequality for functions in the Sobolev space $W^{1,2}$ of a domain in $R^2$. We also deduce a new necessary condition for the complex Monge-Ampere equation for a given measure on a compact Kahler manifold to admit a Holder continuous solution.

Canonical metrics on holomorphic Courant algebroids

The solution of the Calabi Conjecture by Yau implies that every Kähler Calabi–Yau manifold X $X$ admits a metric with holonomy contained in SU ( n ) $\mathrm{SU}(n)$ , and that these metrics are parameterized by the positive cone in H 1 , 1 ( X , R ) $H^{1,1}(X,\mathbb {R})$ . In this work, we give evidence of an extension of Yau's theorem to non-Kähler manifolds, where X $X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid Q $Q$ of Bott–Chern type. The equations that define our notion of best metric correspond to a mild generalization of the Hull–Strominger system, whereas the role of H 1 , 1 ( X , R ) $H^{1,1}(X,\mathbb {R})$ is played by an affine space of ‘Aeppli classes’ naturally associated to Q $Q$ via Bott–Chern secondary characteristic classes.

Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds

Let $(X,\omega)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Ampere equations with right-hand side in $L^p$, $p>1$. Using this we prove that the solutions are Holder continuous with the same exponent as in the Kahler case \cite{DDGKPZ14}. Our techniques also apply to the setting of big cohomology classes on compact Kahler manifolds.

Regularity for a Class of Singular Complex Hessian Equations

In this paper, we study the regularity of the complex Hessian equation when the right hand has pole singularity. We show the Holder continuity of the solution to the Dirichlet problem. In particular, for the complex Monge-Ampere equation, we improve a result of [7].

The quaternionic Calabi Conjecture on abelian Hypercomplex Nilmanifolds Viewed as Tori Fibrations

Abstract We study the quaternionic Calabi–Yau problem in HyperKähler manifolds with torsion geometry, introduced by Alesker and Verbitsky in [ 5], on eight-dimensional two-step nilmanifolds $M$ with an Abelian hypercomplex structure. We show that on these manifolds the quaternionic Monge–Ampère equation can always be solved for any data that are invariant under the action of a three-torus.

Some interior regularity estimates for solutions of complex Monge–Ampère equations on a ball

In this paper, we consider the Dirichlet problem of a complex Monge–Ampère equation on a ball in \({\mathbb {C}}^n\). With \({\mathcal {C}}^{1,\alpha }\) (resp. \({\mathcal {C}}^{0,\alpha }\)) data, we prove an interior \({\mathcal {C}}^{1,\alpha }\) (resp. \({\mathcal {C}}^{0,\alpha }\)) estimate for the solution. These estimates are generalized versions of the Bedford–Taylor interior \({\mathcal {C}}^{1,1}\) estimate.

Schauder estimates for equations with cone metrics, I

This is the first paper in a series to develop a linear and nonlinear theory for elliptic and parabolic equations on Kahler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex Monge-Ampere equations when the background Kahler metrics on $\mathbb{C}^n$ have cone singularities along a smooth complex hypersurface. We prove a sharp pointwise Schauder estimate for linear elliptic and parabolic equations on $\mathbb{C}^n$ with background metric $g_\beta= \sqrt{-1} ( dz_1 \wedge d\bar{z_1} + \ldots + \beta^2|z_n|^{-2(1-\beta)} dz_n \wedge d\bar{z_n}) $ for $\beta\in (0,1)$. Our results give an effective elliptic Schauder estimate of Donaldson and a direct proof for the short time existence of the conical Kahler-Ricci flow.

Bounded strictly pseudoconvex domains in C2 with obstruction flat boundary

On a bounded strictly pseudoconvex domain in $\Bbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local curvature invariant of the boundary. For bounded strictly pseudoconvex domains in $\Bbb{C}^2$ which are diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. In particular we observe that, up to biholomorphism, the unit ball in $\Bbb{C}^2$ is rigid with respect to deformations in the class of strictly pseudoconvex domains with obstruction flat boundary. We further show that for more general deformations of the unit ball, the order of vanishing of the obstruction equals the order of vanishing of the CR curvature. Finally, we give a generalization of the recent result of the second author that for an abstract CR manifold with transverse symmetry, obstruction flatness implies local equivalence to the CR $3$-sphere.

The Pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flows

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Ampere equations, modelled on the Kahler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural assumptions on the Cauchy-Dirichlet boundary data, we show that the envelope of pluripotential subsolutions is semi-concave in time and continuous in space, and provides the unique pluripotential solution with such regularity.

New proof for the regularity of Monge–Ampère type equations

By employing the Green function, in this paper we provide a new and elementary proof for the interior regularity of solutions to the Monge–Ampere equation. This proof also applies to the complex Monge–Ampere equation and the $k$-Hessian equation.

A Priori Estimate for the Complex Monge–Ampère Equation

In this paper, we use the Sobolev type inequality in Wang et al. (Moser–Trudinger inequality for the complex Monge–Ampere equation, arXiv:2003.06056v1 (2020)) to establish the uniform estimate and the Holder continuity for solutions to the complex Monge–Ampere equation with the right-hand side in $$L^p$$ for any given $$p>1$$ . Our proof uses various PDE techniques but not the pluripotential theory.

Some results on the complex Monge-Ampère equation

The complex Monge-Ampere equation has been the subject ofintensive studies in the past fifty years, since its significant applications in complex analysis, complex geometry and Kahler geometry. In this paper, we will introduce our recent work on the Liouville type theorem, $C^{1,~\\alpha}$ and $C^{2,~\\alpha~}$ estimates of the complex Monge-Ampere equation and its applications in Sasakian geometry. These works are joint with S. Dinew, P. F. Guan, C. Li, J. Y. Li and X. W. Zhang.

Special metrics in Kähler geometry

In this note we walk through some of the developments and contributions made in the last 50 years to the study of the complex Monge–Ampere equation in relation to the Calabi conjecture and its generalization.

Log-concavity of volume and complex Monge–Ampère equations with prescribed singularity

Let $$(X,\omega )$$ be a compact Kahler manifold. We prove the existence and uniqueness of solutions to complex Monge–Ampere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the Kahler case as well. As an application we confirm a conjecture by Boucksom–Eyssidieux–Guedj–Zeriahi concerning log-concavity of the volume of closed positive (1, 1)-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn–Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and P-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.

Monge–Ampére iteration

Recently Darvas–Rubinstein proved a convergence result for the Kahler–Ricci iteration, which is a sequence of recursively defined complex Monge–Ampere equations. We introduce the Monge–Ampere iteration to be an analogous, but in a sense more general, sequence of recursively defined real Monge–Ampere second boundary value problems, and we establish sufficient conditions for its convergence. Each step in the iteration is a carefully chosen optimal transportation problem. We determine two cases where the convergence conditions are satisfied and provide geometric applications for both. First, we give a new proof of Darvas and Rubinstein’s general theorem on the convergence of the Ricci iteration in the case of toric Kahler manifolds, while at the same time generalizing their theorem to general convex bodies. Second, we introduce the affine iteration to be a sequence of prescribed affine normal problems and prove its convergence to an affine sphere. These give a new approach to recent existence and uniqueness results due to Berman–Berndtsson and Klartag.

Complex interpolation of $\mathbb R$-norms, duality and foliations

The complex method of interpolation, going back to Calderon and Coifman et al., on the one hand, and the Alexander–Wermer–Slodkowski theorem on polynomial hulls with convex fibers, on the other hand, are generalized to a method of interpolation of real (finite-dimensional) Banach spaces and of convex functions. The underlying duality in this method is given by the Legendre transform. Our results can also be interpreted as new properties of solutions of the homogeneous complex Monge–Ampere equation.

Some new estimates for the complex Monge-Ampère equation

In this paper, we consider the complex Monge-Ampere equation posed on a compact Kahler manifold. We show how to get Lp (p < ∞) and L∞ estimates for the gradient of the solution in terms of the continuity of the right-hand side.

Transverse fully nonlinear equations on Sasakian manifolds and applications

We prove a priori estimates for a class of transverse fully nonlinear equations on Sasakian manifolds and give some geometric applications such as the transversion Calabi-Yau theorem for transverse balanced and (strongly) Gauduchon metrics. We also explain that similar results hold on compact oriented, taut, transverse Hermitian foliated manifold of complex co-dimension n, and give some geometric applications such as the transverse Calabi-Yau theorems for transverse Hermitian and (strongly) Gauduchon metrics.