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 (n−1)-plurisubharmonic functions on Hermitian manifolds. result Solves a specific Dirichlet problem on Hermitian manifolds.
Study on new Monge-Ampère functionals and their variational problems.
problem Existence and uniqueness of solutions for nonlinear eigenvalue problems.
method Introduction of a family of real Monge-Ampère functionals and proving Sobolev type inequalities.
result Existence of solutions for a nonlinear eigenvalue problem.
Solves a general class of free boundary Monge-Ampère equations.
problem Optimal transport with degenerate densities and geometric problems.
method Analyzes a specific class of Monge-Ampère equations and their applications.
result Solves the equations for a general class, including applications to optimal transport and geometric problems.
Proves Hölder continuity of complex Monge-Ampère solutions.
problem Global Hölder continuity of solutions to complex Monge-Ampère equation.
method Analyzes Dirichlet problem on strictly pseudoconvex domains or Hermitian manifolds.
result Proves global Hölder continuity of solutions under given conditions.
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.
Study on Hölder continuity of complex Monge-Ampère solutions on Stein spaces.
problem Understanding continuity of solutions to complex Monge-Ampère equations on Stein spaces.
method Analyzing solutions with Lp densities and Hölder boundary data on Stein spaces with isolated singularities. result Solutions are Hölder continuous outside singular points if boundary data is Hölder continuous.
Researchers solve Dirichlet problem for complex Monge-Ampère equation on Hermitian manifolds.
problem Solving the Dirichlet problem for the complex Monge-Ampère equation on Hermitian manifolds with boundary.
method Weak quasi-plurisubharmonic solutions and optimal subsolution theorems for bounded and Hölder continuous quasi-plurisubharmonic functions.
result Proves continuity of solutions for measures well dominated by capacity, including Lp densities and moderate measures. 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.
In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computa…
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 …
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…
Proposes variational Wasserstein barycenters for geometric clustering.
problem Geometric clustering problems, especially K-means and co-clustering.
method Solves for Monge maps using variational principle, explores connections to K-means and co-clustering.
result Demonstrates feasibility and use of variational Wasserstein barycenters in clustering.
Distributions of Monge type are a class of strongly regular bracket-generating distributions introduced by I. Anderson, Zh. Nie and P. Nurowski. Their symbol algebras prolong to simple graded Lie algebras, thus allowing one to associate a parabolic geometry to any given Monge distribution. This article is devoted to th…
Paper solves a complex equation for unbounded convex sets.
problem Solving the Lp dual Minkowski problem for unbounded closed sets. method Using variational properties of Monge-Ampère functionals, the paper proves existence, regularity, and uniqueness of solutions.
result Existence, regularity, and uniqueness of solutions to the Monge-Ampère type equation for p≥1. Paper solves complex Monge-Ampère equation on almost Hermitian manifolds.
problem Solving Dirichlet problem for complex Monge-Ampère equation.
method Properties of subsolutions for fully nonlinear elliptic equations.
result Existence of C2-smooth strictly J-plurisubharmonic subsolution. Neural networks solve the Dirichlet problem for Monge-Ampère equations.
problem Solving the Dirichlet problem for the Monge-Ampère equation.
method Using deep input convex neural networks to find the unique convex solution.
result Deep input convex neural networks can solve the Monge-Ampère Dirichlet problem.
Solves Dirichlet problem for specific PSH functions on Hermitian manifolds.
problem Solving Dirichlet problem for Monge-Ampère equation for (n−1)-PSH functions. method Deriving a quantitative boundary estimate under (n−1)-PSH subsolutions assumption. result Quantitative boundary estimate confirmed for specific manifolds.
Study solves complex equation on specific types of manifolds.
problem Solving complex Monge-Ampère equation on Kähler manifolds.
method Flow-based arguments to establish existence of smooth solutions.
result Existence of smooth solutions under decreasing right-hand side.
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.
In this paper, we prove a C1,1 estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds. Using this C1,1 estimate, we show existence of C1,1 solutions to the degenerate Monge-Ampère equations, the corresponding Dirichlet problems and the singular Monge-Ampère equatio…
The paper is accompanying "A general Duality Theorem for the Monge-Kantorovich Transport Problem". We explain the methods used in this article in an elementary setting and present two examples complementing the results obtained therein.
We study the problem of the existence and the holomorphicity of the Monge-Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a …
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R^3, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove…
Optimal transport adapted for contaminated probabilities, showing equivalence under specific conditions.
problem Adapting optimal transport for ε-contaminated sets. method Generalized optimal transport problems with lower probabilities, showing equivalence under ε-contaminations. result Monge's and Kantorovich's problems coincide under ε-contaminated sets, but not always. We solve the Dirichlet problem for the complex Monge-Ampère equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Ampère measure of a Hölder continuous plurisubharmonic function. If the boundary data is continuous, then the solution is continuous. If…
Proves existence of eigenvalue and eigenfunction for complex Monge-Ampère operator.
problem Eigenvalue problem for complex Monge-Ampère operator on bounded domains.
method Follows P.L. Lions' strategy for real case, proves new existence theorem for complex degenerate equations, uses a priori estimates and variational approach.
result Existence of first eigenvalue and eigenfunction with specified properties.
Existence and uniqueness of bounded solutions to complex Monge-Ampère flows on Kähler manifolds.
problem Existence and uniqueness of bounded solutions to complex Monge-Ampère flows.
method Proved existence and uniqueness of bounded solutions with specific conditions on the right-hand side.
result Existence and uniqueness of bounded solutions to the complex Monge-Ampère flow on compact Kähler manifolds.
In this lecture delivered at the Integrable and Quantum Field Theory at Peyresq sixth meeting, we review the Lychagin's Monge-Ampere operators theory and exhibit the link it establishes between the classical problem of local equivalence for non linear partial differential equations and the problem of integrability of s…
Derives formulas for Monge-Ampère measures and reduces complex conjectures to simpler existence problems.
problem Complex Monge-Ampère measures and their applications in algebraic geometry.
method Derives formulas and reduces conjectures to simpler existence problems.
result Reduces uniform Yau-Tian-Donaldson conjecture to existence of approximate decompositions.
Study solves Monge-Ampère equation for complete Calabi-Yau metrics.
problem Existence of complete Calabi-Yau metrics on log Calabi-Yau pairs.
method Free boundary Monge-Ampère equation, Legendre duality, existence proof in smooth and Hölder spaces.
result Existence and strict convexity of solutions in smooth and Hölder spaces.
New classification of hyperbolic Monge-Ampère systems with S1=0.
problem Characterizing hyperbolic Monge-Ampère systems with S1=0. method Analyzing invariant tensors S1 and S2 to classify systems with S1=0. result All S1=0 systems with cohomogeneity at most one are linear up to contact transformations. We present a deep generative model, named Monge-Ampère flow, which builds on continuous-time gradient flow arising from the Monge-Ampère equation in optimal transport theory. The generative map from the latent space to the data space follows a dynamical system, where a learnable potential function guides a compressible…
In a recent paper, 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 more general, sequence of recursively defined real Monge-Ampere second boundary valu…
We study the Dirichlet problem for complex Monge-Ampere equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in the flat case. We also consider the equation on compact manifolds without boundary, attempting to …
Study complex Monge-Ampère flows on Kähler manifolds using Perron method.
problem Complex Monge-Ampère flows in big cohomology classes.
method Perron method for pluripotential subsolutions.
result Upper envelope of subsolutions is a unique pluripotential solution with regularity.
New method learns disentangled representations using Gromov-Monge maps.
problem Learning disentangled representations from unlabelled data.
method Introduces a novel approach based on Gromov-Monge maps to preserve geometric features while aligning data distributions.
result Demonstrates effectiveness on four benchmarks, outperforming other methods.
The paper solves Monge-Ampère equations on reflexive polytopes, linking solvability to SYZ conjecture and tropical geometry.
problem Solvability of Monge-Ampère equations on reflexive polytopes.
method Analyzes reflexive polytopes with height functions, proving conditions for Monge-Ampère solvability and linking to SYZ conjecture.
result Conditions for Monge-Ampère solvability are necessary and sufficient, and solvability implies the SYZ conjecture for Calabi-Yau hypersurfaces.
New insights into mirror symmetry via Monge-Ampère domains and pre-Frobenius manifolds.
problem Exploring mirror symmetry using Landau-Ginzburg models and probability densities.
method Investigating Landau-Ginzburg models through Koopman-von Neumann's construction, showing existence of Monge-Ampère domains, and proving mirror pairs via Berglund-Hubsch-Krawitz construction.
result Existence of Monge-Ampère domains and their connection to pre-Frobenius manifolds.
In this paper we consider three deeply connected classificational problems on four-dimensional manifolds. First we consider and describe locally regular distributions. Second we give a classification of almost complex structures of general position in terms of distributions. Finally we classify nondegenerate Monge-Ampe…
Estimates optimal transport maps with known cost functions.
problem Ensuring optimal transport maps correspond to real-world usefulness.
method Differentiable neural ground costs with known Monge map forms.
result General approach for incorporating prior information.
In this paper, by the method of moving planes, we prove the symmetry result which says that classical solutions of Monge-Ampere system in the whole plane are symmetric about some point. Our system under consideration comes from the differential geometry problem.
The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,α} estimates away from a divisor are obtained, by combining techniques of Blocki, Tsuji, Yau, and pluripotential theory. In particular, C^{1,α} geod…
New theorem connects probabilistic permanental point processes to Monge-Ampère equation.
problem Probabilistic interpretation of Monge-Ampère equation boundary value problem.
method Large deviation principles and optimal transport theory.
result Explicit rate function for permanental point processes large deviation.
Study estimates eigenvalues for concave Hessian operators on convex domains.
problem Estimating eigenvalues for concave elliptic Hessian operators.
method Investigates Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators.
result Existence and properties of the first nonzero eigenvalue and eigenfunction.
Study variational problems in Kähler geometry to construct metrics.
problem Maximizing/minimizing Monge--Ampère energy on Kähler potentials.
method Prove existence and uniqueness of extremals, use them to construct metrics.
result Existence and uniqueness of extremals with simple characterization.
We survey the Dirichlet problem for the complex Homogeneous Monge-Ampère Equation, both in the case of domains in Cn and the case of compact Kähler manifolds parametrized by a Riemann surface with boundary. We then give a self-contained account of previous work of the authors that connects this with the Hele…
This work introduces methods to compute optimal Monge maps and learn elastic costs for efficient data mapping.
problem Efficiently mapping one probability distribution to another using elastic costs.
method Proposes numerical methods to compute optimal Monge maps and a learning loss for parameterized regularizers.
result Proves the optimality of computed Monge maps and learns the parameters of elastic costs.
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.