arXiv research

Rolling two hyperboloid surfaces is described using a Monge normal form.

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

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

We find normal forms for parabolic Monge-Ampere equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the analytic case, this equation is shown to have solutions. The other normal forms exha…

Study bounds on Monge-Ampère volumes for degenerate complex equations.

We define a non-degenerated Monge-Ampere structure on a 6-manifold associated with a Monge-Ampere equation as a couple (Ω,ω), such that Ωis a symplectic form and ωis a 3-differential form which satisfies ω\wedgeΩ=0 and which is non-degenerated in the sense of Hitchin. We associate with such a couple an almost (pseudo) …

Study on $m$-positivity in Kähler manifolds with new Monge-Ampère-type equation.

New geometries derived from symplectic Monge-Ampère structures.

Unconditional proof of Demailly's transcendental Morse inequality for higher cohomology classes using a general gauge-fixing for the Monge-Ampère-type equation.

We complete the list of normal forms for effective 3-forms with constant coefficients with respect to the natural action of symplectomorphisms in \mathbb{R}^6. We show that the 3-form which corresponds to the Special Lagrangian equation is among the new members of the classification. The symplectic symmetry algebras an…

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

Proves a Liouville theorem for complex Monge-Ampère equation on product manifolds.

Classifies jets of surfaces in 4-space based on their projections.

Classifies integrable systems related to geometry.

We define a nondegenerate Monge-Ampère structure on a 6-dimensional manifold as a pair $(Ω,ω)$, such that $Ω$ is a symplectic form and $ω$ is a 3-differential form which satisfies $ω\wedgeΩ=0$ and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi-Yau stru…

Uniform bounds for Green's function on Kähler manifolds derived from complex Monge-Ampère equations.

This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge-Ampère equations, by using the so-called "meta-symplectic structure" ass…

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

Solves complex Monge-Ampère equations with Hölder continuous solutions in Kähler manifolds.

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations …

Auxiliary equations improve bounds in symplectic geometry.

We prove an existence result for a "generalised" Monge-Ampère equation introduced earlier under some assumptions on a flat complex 3-torus. As an application we prove the existence of Chern connections on certain kinds of holomorphic vector bundles on complex 3-tori whose top Chern character forms are given representat…

Proves existence and uniqueness of solutions to a quaternionic Monge-Ampère equation.

Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.

Introduces a new equation for complex surfaces, proving stability and inequalities.

Estimates optimal transport maps with known cost functions.

Solves second boundary value problem for affine mean curvature equation and related Monge-Ampère equations.

Existence and boundary regularity away from the corners are established for two-dimensional Monge-Ampère equations on convex polytopes with Guillemin boundary conditions. An important step is to derive an expansion in terms of functions $y^n$ and $y^n\log y$ for solutions to equations of the form $\det D^2u(x,y) = y^{-…

The classes of Monge-Ampère systems, decomposable and bi-decomposable Monge-Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of …

Introduces a new PDE involving differential forms for Kähler geometry.

Uniform estimates for complex equations on compact manifolds found.

Study of symplectic Monge-Ampère equations using moment maps and contact structures.

We show existence of unique smooth solutions to the Monge-Ampere equation for (n-1)-plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As a consequence we obtain Calabi-Yau theorems for Gauduchon and strongly Gauduchon metrics on a class of non-Kahler manifolds: those satisfyi…

Researchers classify and characterize Bäcklund transformations for hyperbolic Monge-Ampère systems.

Defines volume and Monge-Ampère energy on polarized affine varieties.

Let $X$ be a compact Kähler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_χ(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $χ$ has fast growth at infinity, the corresponding functions are …

Let u be a function of n independent variables x^1, ..., x^n, and U=(u_{ij}) the Hessian matrix of u. The symplectic Monge-Ampere equation is defined as a linear relation among all possible minors of U. Particular examples include the equation det U=1 governing improper affine spheres and the so-called heavenly equatio…

Study local topological types of binary differential equations for surface families.

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibilit…

Paper develops techniques to solve complex PDEs involving higher cohomology forms.

Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature

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 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 symplectic 4-manifold theory.

Study of Monge-Ampère volumes on hermitian manifolds, focusing on plurisigned metrics.

Study confirms geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.

Study estimates eigenvalues for concave Hessian operators on convex domains.

Mathematically, a homothetic function is a function of the form $f({\bf x})=F(h(x_1,...,x_n))$, where $h$ is a homogeneous function of any degree $d\ne 0$ and $F$ is a monotonically increasing function. In economics homothetic functions are production functions whose marginal technical rate of substitution is homogeneo…

We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, w…