We give necessary and sufficient conditions for existence of solutions to a general system of complex Monge-Ampère equations on Fano horosymmetric manifolds. In particular, we get necessary and sufficient conditions for existence of coupled Kähler-Ricci solitons, Mabuchi metrics and twisted Kähler-Einstein metrics in t…
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48 results for “Monge coupling”
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) …
Develop a framework for barycentric projections of optimal transport plans on Riemannian manifolds.
problem Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps.
method Develop a framework for barycentric projections of transport couplings on Riemannian manifolds.
result The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss.
Two probability distributions and in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are al…
New bounds on optimal transport regularization show faster convergence rates than previously known.
problem Understanding the localization rate of Quadratically Regularized Optimal Transport (QOT) optimizers.
method Established lower bounds and derived mean-squared deviation controls for QOT optimizers.
result Lower bound of support concentration rate in directed Hausdorff distance.
Uniform bounds derived for fully non-linear equations.
problem Bounding fully non-linear equations uniformly in background metrics.
method Auxiliary Monge-Ampère equations and entropy-like quantities.
result Uniform bounds for systems coupling fully non-linear equations to their linearizations.
New optimal transport divergences derived from scoring functions.
problem Developing new divergences for optimal transport.
method Using scoring functions as cost functions in optimal transport.
result Comonotonic coupling is optimal for many new divergences.
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.
These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Ampère and linearized Monge-Ampère equations. The prescribed affine mean curvature equation is a fully nonlinear, fourth order, geometri…
Unified approach solves Kyle model with dynamic information.
problem Solving a generalized Kyle model with dynamic information.
method Monge-Kantorovich duality and backward stochastic partial differential equations.
result Characterization of optimal strategies and pricing rules.
The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal tran…
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.
Estimation of Monge Matricesmath.ST
Monge matrices and their permuted versions known as pre-Monge matrices naturally appear in many domains across science and engineering. While the rich structural properties of such matrices have long been leveraged for algorithmic purposes, little is known about their impact on statistical estimation. In this work, we …
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.
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.
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.
Coupling between Phase Separation and Geometry on a Closed Elastic Curve: Free Energy Minimization and Dynamicscond-mat.soft
Study of phase separation and geometry on a closed elastic curve, including dynamics and free energy minimization.
problem Free energy and dynamics of a closed elastic filament coupled to a scalar concentration field.
method Analytical and numerical simulations of coupled Willmore flow and Cahn--Hilliard gradient flow on differential geometry.
result Qualitative changes in free energy landscape due to closure constraint, leading to metastable and stable multi-domain morphologies.
Solves complex Monge-Ampère equations on Kähler manifolds.
problem Behavior of singularities in solutions to degenerate equations.
method Analyzes singularities of solutions to degenerate complex Monge-Ampère equations.
result Resolves unresolved problem from Yau's work.
Monge-Ampère type equation for the Nakano positive curvature tensor of holomorphic vector bundlesmath.DG
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…
Study compares Monge-Ampère capacities on Kähler manifolds.
problem Comparing Monge-Ampère capacities on compact Kähler manifolds.
method Proved all capacities comparable, used Xia's integration by parts formula.
result All Monge-Ampère capacities are comparable.
The paper studies quaternionic Monge-Ampère equations in weighted energy classes.
problem Characterizing the finite energy range of quaternionic Monge-Ampère operator.
method Proving well-definedness and fine property of the operator in weighted energy classes.
result Explicit characterization of the finite energy range of quaternionic Monge-Ampère operator.
Paper proves uniform continuity bounds for complex Monge-Ampère solutions.
problem Estimating the continuity of solutions to complex Monge-Ampère equations.
method PDE-based approach from fully non-linear equations in Kähler geometry.
result Uniform and sharp estimate for the modulus of continuity.
Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.
problem Complex Monge-Ampère equation with positive Radon measure on compact Hermitian manifolds.
method Proves bounded subsolution theorem.
result Establishes bounded subsolution theorem for complex Monge-Ampère equation.
Second-order generalized Monge--Ampère equations on a plane and its geometric singular solutionsmath.DG
Study geometric singular solutions of generalized Monge-Ampère equations.
problem Solving generalized Monge-Ampère equations on a plane.
method Using exterior differential systems and Cauchy characteristics.
result Criteria for geometric singular solutions to be equivalent to specific types.
Study on residual Monge-Ampère mass of complex functions with directional Lipschitz continuity.
problem Estimating the residual Monge-Ampère mass of plurisubharmonic functions.
method General decomposition formula under Sasakian structure, -apriori estimate, upper-bound estimate on residual mass.
result Upper-bound estimate on residual mass for uniformly directional Lipschitz continuity confirmed.
We study various capacities on compact Kähler manifolds which generalize the Bedford-Taylor Monge-Ampère capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Ampère equations.
New proof for convex solutions of Monge-Ampère equation.
problem Interior regularity of strictly convex solutions
method Doubling inequality for Hessian in extrinsic distance function
result Interior regularity established
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.
Rolling two hyperboloid surfaces is described using a Monge normal form.
problem Describing the rolling motion of hyperboloid surfaces.
method Parametrization of using unimodular fractional linear transformations.
result Found a Monge normal form for the rolling of hyperboloid surfaces.
The paper proves a comparison principle for complex Monge-Ampère flows and solves a uniqueness problem.
problem Proving uniqueness of weak solutions to the pluripotential Cauchy-Dirichlet problem.
method Proving a comparison principle for the pluripotential complex Monge-Ampère flows.
result Proves the uniqueness of the weak solution to the pluripotential Cauchy-Dirichlet problem.
Note on gradient estimates for complex Monge-Ampere equation.
problem Gradient estimates for solutions of complex Monge-Ampere equation.
method Estimates and for gradient in terms of continuity of the right-hand side.
result Gradient estimates for solutions of complex Monge-Ampere equation.
New proof of estimates for Monge-Ampère and Hessian equations on nef classes.
problem Estimating solutions to Monge-Ampère and Hessian equations on nef classes.
method Applying PDE approach to Kähler manifolds to nef classes.
result New proofs of estimates for Monge-Ampère and Hessian equations.
Paper establishes estimates for complex Monge-Ampere and Hessian equations.
problem Estimating solutions to complex Monge-Ampere and Hessian equations.
method Uses PDE techniques similar to Phong et al to prove and Hölder estimates.
result Establishes estimates for both complex Monge-Ampere and Hessian equations.
We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere equation.
Study complex Monge-Ampère equations on compact Kähler manifolds.
problem Finite energy range of complex Monge-Ampère operator.
method Survey and general answer to Guedj-Zeriahi's question.
result General answer to Guedj-Zeriahi's question about finite energy range.
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.
Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.
problem Existence and uniqueness of solutions to a generalized Monge-Ampère equation.
method Analyzes the equation's dependence on almost Kähler structure and proves Donaldson's conjecture.
result Proves Donaldson's conjecture for tamed almost complex 4-manifolds.
We investigate a class of multi-dimensional two-component systems of Monge-Ampère type that can be viewed as generalisations of heavenly-type equations appearing in self-dual Ricci-flat geometry. Based on the Jordan-Kronecker theory of skew-symmetric matrix pencils, a classification of normal forms of such systems is o…
Enhances gradient estimates for Hermitian Monge-Ampère equations.
problem Improving estimates for Hermitian Monge-Ampère equations.
method Improves gradient estimates using Evans-Krylov and third derivatives estimates.
result Enhanced estimates for second and third order derivatives.
Solves a specific Dirichlet problem on Hermitian manifolds.
problem Solving Dirichlet problem for Monge-Ampère type equations on Hermitian manifolds.
method Solves the Dirichlet problem for Monge-Ampère type equations for -plurisubharmonic functions on Hermitian manifolds.
result Solves a specific Dirichlet problem on Hermitian manifolds.
In this note, we describe the Hermitian metrics that leave the total Monge-Ampere volume invariant. In particular, we give several characterizations of the Hermitian metrics which satisfy the comparison principle for the complex Monge-Ampere operator
Alternative proof of a theorem using parabolic Monge-Ampère equation in HKT geometry.
problem Proving a theorem about solutions to the quaternionic Monge-Ampère equation.
method Generalizing the parabolic Monge-Ampère equation to HKT geometry and proving existence and convergence of solutions.
result Existence and convergence of solutions to the equation under certain conditions.