Narrow Results

Using the Calabi–Yau technique to solve Monge-Ampère equations, we translate a result of T. Fujita on approximate Zariski decompositions into an analytic setting and combine this to the holomorphic Morse inequalities in order to express the volume of a line bundle as the maximum of the mean curvatures of all the singular Hermitian metrics on it, with a way to pick an element at which the maximum is reached and satisfying a singular Monge–Ampère equation. This enables us to introduce the volume of any (1,1)-class on a compact Kähler manifold, and Fujita's theorem is then extended to this context.

We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also show the results for surfaces of constant Gaussian curvature and for developable surfaces. In particular we confirm that generic singularities appearing in such a surface are just cuspidal edges and swallowtails.

We show that an n-dimensional real ellipsoid in ℝⁿ⁺¹ with the induced Riemannian metric does not admit an unbounded adapted complexification in the sense of Lempert/Szőke and Guillemin/Stenzel, unless it is a round sphere. In other words, an ellipsoid whose (maximal) Grauert tube has infinite radius must be a round sphere. For the proof we take advantage of the integrability of the geodesic flow and use a classical theorem on umbilic geodesics. We carry out an extension of this result to Liouville metrics elsewhere.

We prove that every locally pluripolar set on a compact complex manifold is pluripolar. This extends similar results in the Kähler case.

We focus on the structure of the solution set for the nonlinear equation

We prove that the optimal transportation mapping that takes a Gaussian measure γ on an infinite dimensional space to an equivalent probability measure g·γ satisfies the Monge–Ampère equation provided that log g∈L¹(γ) and g log g∈L¹(γ).

In this note, we obtain existence results for complete Ricci-flat Kähler metrics on crepant resolutions of singularities of Calabi–Yau varieties. Furthermore, for certain asymptotically flat Calabi–Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial variety.

In this paper, we consider generalized solutions to the Dirichlet problem for a class of generalized Monge–Ampère equations. For such generalized solutions, we give a complete proof for the so-called comparison principle.

In this paper, a cascadic Newton’s method is designed to solve the Monge–Ampère equation. In the process of implementing the cascadic multigrid, we use the Full-Local type interpolation as prolongation operator and Newton iteration as smoother. In order to obtain Full-Local type interpolation, we provide several finite difference stencils. Especially, the skewed finite difference methods are first applied by us for the elliptic Monge–Ampère equation. Based on Full-Local interpolation techniques and cascade principle, the new algorithm can save a large amount of computation time. Some numerical experiments are provided to confirm the efficiency of our proposed method.

We apply the Wick rotation to the Monge–Ampère equation of Tzitzeica graphs and we introduce the Wick–Tzitzeica solitons as complex functions solving the new equation. Some known Tzitzeica surfaces yields examples of these new solitons and we analyze them. To a second Wick–Tzitzeica soliton, we associate a homogeneous ODE system of gradient type which is Nambu–Poisson.

Consider two bounded domains Ω and Λ in ℝ², and two sufficiently regular probability measures μ and ν supported on them. By Brenier's theorem, there exists a unique transportation map T satisfying T_#μ = ν and minimizing the quadratic cost ∫_ℝⁿ ∣T(x) - x∣² dμ(x). Furthermore, by Caffarelli's regularity theory for the real Monge–Ampère equation, if Λ is convex, T is continuous.

We study the reverse problem, namely, when is T discontinuous if Λ fails to be convex? We prove a result guaranteeing the discontinuity of T in terms of the geometries of Λ and Ω in the two-dimensional case. The main idea is to use tools of convex analysis and the extrinsic geometry of ∂Λ to distinguish between Brenier and Alexandrov weak solutions of the Monge–Ampère equation. We also use this approach to give a new proof of a result due to Wolfson and Urbas.

We conclude by revisiting an example of Caffarelli, giving a detailed study of a discontinuous map between two explicit domains, and determining precisely where the discontinuities occur.

The solution of the Strominger system can be viewed as a canonical structure on non-Kähler Calabi-Yau threefolds with balanced metrics. In this talk, we review the existence of balanced metrics on non-Kähler complex manifolds and the existence of solutions to the Strominger system.