The study identifies Hermitian metrics preserving the total Monge-Ampere volume.
problem Understanding Hermitian metrics preserving volume invariance.
method Characterizations and comparison principles for complex Monge-Ampere operator.
result Several characterizations of Hermitian metrics satisfying the comparison principle.
Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.
problem Volume noncollapsing for Kähler metrics induced by complex Monge-Ampère equations.
method Proves local volume noncollapsing estimate with Ricci curvature lower bound.
result Establishes diameter and gradient estimates for Kähler metrics.
Study bounds on Monge-Ampère volumes for degenerate complex equations.
problem Bounds on volumes of Monge-Ampère measures for degenerate complex equations.
method Fine use of quasi-plurisubharmonic envelopes.
result Established a transcendental version of the Grauert-Riemenschneider conjecture.
The paper studies volumes of direct images for high tensor powers of ample bundles.
problem Understanding asymptotics of Monge-Ampère volumes for high tensor powers of ample line bundles.
method Analyzes the leading term of asymptotics and classifies bundles saturating a topological bound.
result Provides a characterization of bundles admitting projectively flat Hermitian structures in the case of high symmetric powers of ample vector bundles.
Defines volume and Monge-Ampère energy on polarized affine varieties.
problem Volume and Monge-Ampère energy on polarized affine varieties.
method Definition of volume using asymptotics of jumping numbers, Monge-Ampère energy using forms and currents on Berkovich spaces.
result Monge-Ampère energy agrees with volume of filtrations and recovers known functionals.
Study volumes of Bott-Chern classes on complex manifolds.
problem Understanding volumes of transcendental Bott-Chern classes.
method Extending non-pluripolar products to quasi-positive currents, establishing quasi-monotonicity of Monge-Ampère masses, and solving degenerate complex Monge-Ampère equations.
result Positive answer to Demailly-Păun-Boucksom conjecture regarding bounded mass property.
New families of Ricci solitons found with collapsing volume.
problem Finding new Ricci solitons with specific volume behavior.
method Reduced soliton equation to Monge-Ampère equation coupled with ODEs.
result Explicit complete expanding solitons and existence results for other types.
The paper proves complex Monge-Ampère equations with new singularity types and confirms log-concavity of volume.
problem Existence and uniqueness of solutions to complex Monge-Ampère equations with prescribed singularities.
method Proves existence and uniqueness of solutions to complex Monge-Ampère equations with general model type singularities in big cohomology classes.
result Log-concavity of volume of closed positive (1,1)-currents is confirmed.
Study of Monge-Ampère volumes on hermitian manifolds, focusing on plurisigned metrics.
problem Behavior of Monge-Ampère volumes under hermitian variations.
method Analysis of hermitian forms and cohomology, studying boundedness and positivity.
result Existence and non-existence of plurisigned hermitian metrics on specific manifolds.
We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to Monge-Ampere Equations.
Study shows uniqueness of solutions on complex manifolds without requiring solution decay.
problem Uniqueness of solutions to Monge-Ampere equation on complex manifolds.
method Caccioppoli inequality techniques applied to Kähler manifolds with sub-quadratic volume growth.
result Uniqueness of bounded C1,1 solutions to Monge-Ampere equation without decay requirement. Uniform bounds for Green's function on Kähler manifolds derived from complex Monge-Ampère equations.
problem Uniform bounds for Green's function on Kähler manifolds.
method Auxiliary Monge-Ampère equations, non-linear proof.
result Uniform lower bounds for the Green's function on Kähler manifolds.
This paper proves Hölder continuity for complex Monge-Ampère equations on Kähler varieties.
problem Establishing Hölder estimates on singular Kähler varieties.
method Geometric regularization based on partial C0 estimate. result Uniform Hölder continuity for complex Monge-Ampère equations on Kähler varieties.
New method solves complex Monge-Ampère equations on hermitian manifolds.
problem Solving degenerate complex Monge-Ampère equations on hermitian manifolds.
method New approach using compactness and envelopes properties of quasi-plurisubharmonic functions.
result New relative a priori estimates and existence results for degenerate complex Monge-Ampère equations.
Paper solves Monge-Ampère equations on compact Hessian manifolds using optimal transport.
problem Solving Monge-Ampère equations on compact Hessian manifolds with periodicity constraints.
method Variational framework involving dual manifolds and Legendre transform.
result Existence and uniqueness results for Monge-Ampère equations.
Researchers classify and decompose valuations on convex functions.
problem Classifying valuations on convex functions.
method Geometric decomposition of valuations, using properties of special subspaces and Monge-Ampère-type operators.
result Valuations decompose into subspaces defined by vanishing properties.
We prove a conjecture saying that complex projective space has maximal volume (degree) among all toric Kaehler-Einstein manifolds of dimension n. The proof is inspired by our recent work on sharp Moser-Trudinger and Brezis-Merle type inequalities for the complex Monge-Ampere operator, but is essentially self-contained.
A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kaehler covering W, with the deck transform acting on W by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its …
Estimates Kaehler metrics' diameter in big cohomology classes.
problem Estimating the diameter of Kaehler metrics in big cohomology classes.
method Proves uniform diameter estimates using integrability conditions and stability properties of complex Monge-Ampere equations.
result Uniform diameter estimates for Kaehler metrics in big cohomology classes.
The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.
problem Investigating volume invariants for Hermitian-symplectic metrics.
method Introducing a functional acting on metrics in Aeppli cohomology classes and proving critical points are Kähler.
result The volume invariant generalises the volume of a Kähler class and vanishing is a necessary condition for the existence of a Kähler metric.
Characterizes closures of test configurations and algebraic singularity types.
problem Understanding closures of test configurations and algebraic singularity types.
method Analyzes metric spaces of L1 geodesic rays and characterizes closures of singularity types. result Arithmetic and non-pluripolar volumes coincide for algebraic singularity types, and equality holds on their closure.
Researchers find Kähler-Einstein metrics near isolated log terminal singularities.
problem Existence of Kähler-Einstein metrics with positive curvature near isolated log terminal singularities.
method Solving complex Monge-Ampère equations to analyze the existence of metrics.
result Existence of smooth solutions in subcritical regimes, with critical exponent expressed in terms of normalized volume.
Estimates Kähler metric diameters with entropy bound alone.
problem Estimating Kähler metric diameters.
method PDE techniques for L∞ estimates of the Monge-Ampère equation, improving degeneracies. result Diameter bounds for Kähler-Ricci flow and Calabi-Yau manifolds.
Researchers find a way to estimate potential functions for quaternionic metrics.
problem Existence of quaternionic Gauduchon metrics with prescribed volume form.
method Reframed as a fully nonlinear elliptic equation and established a uniform estimate.
result Uniform estimate for the potential function.
Researchers solve Calabi-Yau equation on Kodaira-Thurston manifold.
problem Solving the Calabi-Yau equation on a specific manifold.
method Using an invariant almost-Kaehler structure and an ansatz to reduce the problem to a Monge-Ampère equation.
result The problem is reduced to a Monge-Ampère equation under certain conditions.
New conical metrics found on toric varieties with convex cones.
problem Finding conical metrics on toric affine varieties.
method Existence result for inhomogeneous Monge-Ampere equation, transversal a priori estimates.
result Existence of conical Ricci flat Kahler metrics on Q-Gorenstein affine toric varieties.
Local solubility of Bao--Ratiu equations proven for surfaces with specific curvature conditions.
problem Existence of asymptotic directions for volume-preserving diffeomorphisms on surfaces.
method Analysis of degenerate Monge--Ampère equation following Han's work.
result Asymptotic directions always exist locally about a point on surfaces with specific curvature conditions.
Study shows how to calculate the volume of pseudoeffective line bundles on Kähler manifolds.
problem Calculating the volume of pseudoeffective line bundles on Kähler manifolds.
method Using a limit of section dimensions and a model potential associated to the line bundle.
result The limit of k−nh0(X,Lk⊗I(ku)) equals the non-pluripolar volume of P[u]I. New geometries derived from symplectic Monge-Ampère structures.
problem Exploring new generalized geometries from symplectic Monge-Ampère structures.
method Inspired by Hu, Moraru, and Svoboda, constructing new geometries from non-degenerate 2D symplectic Monge-Ampère structures.
result Non-degenerate Monge-Ampère structures give rise to quadric surfaces of generalized almost geometries.
The paper generalizes Monge-Ampère equations and their solutions in differential geometry.
problem Understanding the structure of Monge-Ampère equations and their solutions.
method Generalizing Monge-Ampère equations to higher-order systems and proving their solutions correspond to integral manifolds of exterior differential systems.
result The Korteweg-de Vries (KdV) equation and Cauchy-Riemann equations are examples of generalized Monge-Ampère equations.
General existence theorem for complex Monge-Ampère on hyperconvex domains.
problem Existence of solutions to complex Monge-Ampère equations on hyperconvex domains.
method General existence theorem.
result Existence of weak solutions to complex Monge-Ampère equations on hyperconvex domains.
Study thermodynamic framework for Monge-Ampère equations on real tori.
problem Monge-Ampère equations on real tori.
method Thermodynamic framework, point processes, convergence in law.
result Convergence in law of point processes associated with Monge-Ampère equations.
Existence theorem for complex Monge-Ampère on compact Kähler manifolds.
problem Existence of solutions to complex Monge-Ampère equations on compact Kähler manifolds.
method General existence theorem
result Existence of weak solutions to complex Monge-Ampère equations on compact Kähler manifolds.
Solves complex Monge-Ampère equation for measures with pluripolar parts.
problem Characterizing measures with complex Monge-Ampère equation solutions.
method Solves for measures with a pluripolar part in compact Kähler manifolds.
result Generalizes classical results in bounded hyperconvex domains.
New estimate for complex Monge-Ampère equations improves previous results.
problem Improving estimates for complex Monge-Ampère equations.
method Using the ABP maximum principle to prove a new gradient estimate.
result Proves a new gradient estimate for complex Monge-Ampère equations.
Solves Monge-Ampère on non-integrable manifolds.
problem Existence and uniqueness of solutions to Monge-Ampère equation on non-integrable almost complex manifolds.
method Existence and uniqueness proved using the Monge-Ampère equation.
result Existence and uniqueness of solutions proved.
Unique solution found for quaternionic Monge-Ampère equation on specific HKT manifolds.
problem Solving the quaternionic Monge-Ampère equation on HKT manifolds with an HKT foliation.
method Study of quaternionic Monge-Ampère equation on HKT manifolds with specific foliation properties.
result Unique solution for the equation for every basic datum.
Study proves long-term solutions to a specific equation on hyperKähler manifolds.
problem Proving long-term existence and uniqueness of solutions to a parabolic quaternionic Monge-Ampère equation.
method Proved long-term existence and uniqueness using parabolic quaternionic Monge-Ampère type equation.
result Solution converges smoothly to the unique solution of the Monge-Ampère equation.
Paper proves C1,1 regularity for complex Monge-Ampère equations.
problem Complex Monge-Ampère equations on compact almost Hermitian manifolds.
method Proves C1,1 estimate and uses it to show existence of solutions. result Proves C1,1 regularity for geodesics in Sasakian metrics. Gradient and Laplacian estimates for complex Monge-Ampère equations found.
problem Estimating solutions to complex Monge-Ampère equations with singularities.
method Integral method applied to obtain gradient and Laplacian estimates.
result Gradient and Laplacian estimates for the solution to the singular complex Monge-Ampère equation.
Smooth solutions found for complex Monge-Ampère flows.
problem Regulating smoothness in complex Monge-Ampère flows.
method Analyzing initial ω-psh functions with zero Lelong number. result General Monge-Ampère flow solution is immediately smooth.
Researchers compute limits of Kähler-Einstein forms on degenerating manifolds.
problem Understanding limits of Kähler-Einstein forms on degenerating manifolds.
method Hybrid convergence of Kähler-Einstein measures using algebro-geometric limits.
result Limit measure is a weighted sum of Dirac masses at divisorial valuations.
Uniform estimates for complex Monge-Ampere equations help understand singular solutions and collapsing metrics.
problem Uniform estimates for complex Monge-Ampere equations to study geometric regularity.
method Proved uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations.
result Uniform estimates for complex Monge-Ampere equations applied to singular solutions and collapsing metrics.
Solves a Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
problem Solving Monge-Ampère type equations for Nakano positive curvature tensors of holomorphic vector bundles.
method Solves the Monge-Ampère type equation in the conformal class of a Nakano positive Hermitian metric.
result Solves the Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
Develops a new parabolic equation for surfaces, proving long-time existence and convergence.
problem Extending elliptic equations to parabolic settings for surfaces.
method Introduces a parabolic analogue of the elliptic split-type Monge-Ampère equation.
result Proves long-time existence and convergence conditions for the new equation.
Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.
problem Solving Monge-Ampère equations on compact Hessian manifolds.
method Perron method, compactness properties of normalized quasi-convex functions, local and global comparison principles for twisted Monge-Ampère operators.
result Solves Monge-Ampère equations involving arbitrary probability measures.
We prove several approximation theorems of the complex Monge-Ampere operator on a compact Kahler manifold. As an application we give a new proof of a recent result of Guedj and Zeriahi on a complete description of the range of the complex Monge-Ampere operator in the class of w-plurisubharmonic functions with vanishing…
Use envelope method to prove existence of complex Monge-Ampère equations on compact Kähler manifolds.
problem Existence of solutions to degenerate complex Monge-Ampère equations on compact Kähler manifolds.
method Classical Perron envelope method
result General existence theorem for degenerate complex Monge-Ampère equations on compact Kähler manifolds