Analyzes geodesics in positive Lagrangian spaces on Riemannian manifolds.
problem Analyzes geodesics in positive Lagrangian spaces on Riemannian manifolds.
method Develops analytic techniques to solve the Dirichlet problem for the Riemannian degenerate special Lagrangian equation.
result Continuous geodesics in the space of positive Lagrangians are governed by the Riemannian DSL.
Develops analytic tools for DSL, a geodesic equation in positive graph Lagrangians.
problem Existence of geodesics in positive Lagrangian space.
method Analytic tools, including space-time Lagrangian angle and calibration measure.
result Existence of solutions to DSL in all branches, with well-defined lengths.
New findings on convexity of special Lagrangian geodesics.
problem Convexity of special Lagrangian geodesics in space-time.
method Space-time coordinate transformation preserving Lagrangian angle, leading to C2 estimate. result Subsolutions in all branches of the degenerate special Lagrangian equation are bi-convex.
Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.
problem Understanding geodesics in spaces of positive Lagrangian submanifolds.
method Cylindrical transform of geodesics, solving elliptic PDEs.
result Geodesics in positive Lagrangian spaces correspond to one-parameter families of special Lagrangian cylinders.
Study Monge-Ampère equations on Calabi-Yau hypersurfaces, proving unique solutions and implications for special Lagrangian fibrations.
problem Existence of special Lagrangian fibrations in Calabi-Yau hypersurfaces.
method Non-Archimedean and tropical Monge-Ampère equations on Berkovich and skeleton spaces, proving uniqueness and deriving solutions.
result Unique solutions to tropical and non-Archimedean Monge-Ampère equations, leading to existence of special Lagrangian fibrations.
The paper constructs special Lagrangian n-folds in arbitrary dimensions.
problem Developing a construction for special Lagrangian n-folds in arbitrary dimensions.
method Reduction of special Lagrangian condition to a quasilinear elliptic system of 2D non-linear Cauchy-Riemann equations.
result The structure and multiplicity of singularities are governed by an associated polynomial.
Study of symplectic manifolds degenerating into singular spaces.
problem Understanding degenerations of symplectic manifolds into singular spaces.
method Topological framework focusing on Lagrangian manifolds and holomorphic membranes.
result Degenerations into singular toric varieties yield exotic Lagrangian tori.
Special Lagrangian submanifolds emerge from K3 surface collapse.
problem Understanding special Lagrangian submanifolds in K3 surface collapse.
method Lifting affine lines to degenerating sequences of special Lagrangian submanifolds.
result Constructing special Lagrangian two-spheres connecting Taub-NUT bubbles.
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
Study Hamiltonian formalisms of degenerate gravity Lagrangians.
problem Hamiltonian analysis of degenerate gravity Lagrangians.
method Employed Dirac-Bergmann constraint algorithm and Gotay-Nester-Hinds algorithm.
result Total Hamiltonian functions and Hamilton's equations derived.
Study deformed Hermitian-Yang-Mills equation via GIT and prove existence of geodesics.
problem Existence and regularity of solutions to deformed Hermitian-Yang-Mills equation.
method Variational approach via infinite dimensional GIT problem, scale estimates, Fourier-Mukai transform.
result Existence of smooth and weak geodesics with C1,α regularity. Geodesic connects Lagrangian graphs over torus in complex space.
problem Existence of geodesic connecting Lagrangian graphs in complex space.
method Formulated as a degenerate elliptic equation, solved via Dirichlet problem.
result Geodesic connecting Lagrangian graphs can be constructed.
Construct special Lagrangian fibrations on abelian varieties using retraction techniques.
problem Constructing special Lagrangian fibrations on abelian varieties.
method Explicit construction using special techniques in non-Archimedean geometry.
result Solved a conjecture of Kontsevich-Soibelman for finite quotients of abelian varieties.
Real analytic solutions found for special Lagrangian equation.
problem Analyzing convex solutions of the special Lagrangian equation.
method Interior regularity established for convex viscosity solutions.
result All convex solutions are real analytic in the interior.
Study special Lagrangian submanifolds with edge singularities using elliptic theory.
problem Characterize the moduli space of deformations of special Lagrangian submanifolds with edge singularities.
method Use elliptic theory for edge-degenerate differential operators on singular manifolds.
result Obtain a general theorem describing the local structure of the moduli space.
Estimates for special Lagrangian curvature equations in critical and convex cases.
problem Interior estimates for special Lagrangian curvature equations.
method Establishes a priori interior curvature and gradient estimates.
result Proves interior curvature and gradient estimates for special Lagrangian curvature equations.
Internal Lagrangians derived from variational principles.
problem Reproducing the principle of stationary action in variational geometry.
method Introducing stationary points of internal Lagrangians, establishing connections with symmetries and conservation laws, and investigating relations between non-degenerate and internal Lagrangians.
result Noether's theorem reformulated in terms of internal Lagrangians.
Study Neumann problem for special Lagrangian type equations.
problem Neumann problem for special Lagrangian type equations.
method Uniform a priori estimates, continuity method, direct proof of boundary double normal derivative estimates.
result Existence result for Neumann problem of special Lagrangian type equations.
Paper develops estimates for Lagrangian phase changes in 2D.
problem Interior estimates for Lagrangian phase changes in 2D.
method Modified doubling technique to handle degenerate Jacobi inequalities.
result Interior Hessian and gradient estimates established for critical phase.
In this paper, we discuss the Lagrangian angles of a family of Lagrangian fibrations moved under mean curvature flow. In the case n=1, the angle function is shown to satisfy a degenerated partial differential equation. We prove that any smooth solution to the equation also corresponds to smooth foliation of curves un…
Paper doubles Hessian estimates for special Lagrangian equation with constraints.
problem Estimating Hessian for special Lagrangian equation under general phase constraints.
method Doubling argument, Alexandrov-type theorems.
result Established Hessian estimates for special Lagrangian equation.
Article proves equivalence of two theorems in two dimensions.
problem None explicitly stated in the abstract.
method Explains equivalence between two theorems.
result Equivalence of Jörgens' Theorem and Fu's Theorem.
Proves Hessian estimates for special Lagrangian equation with new proofs.
problem Interior Hessian estimates for special Lagrangian equation
method Doubling proofs, higher codimension analogue of previous methods
result Higher codimension analogue of gradient estimate for minimal hypersurfaces
Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.
problem Second boundary value problem for special Lagrangian curvature potential equation.
method Method of continuity with a-priori estimate.
result Existence and uniqueness of smooth uniformly convex solutions.
Proves smoothness and estimates for special Lagrangian solutions with semi-convexity.
problem Smoothness and estimates for special Lagrangian solutions.
method Viscosity solutions, smoothness, interior derivative estimates, sharpness of conditions.
result New Liouville theorem and effective Hessian estimates for special Lagrangian solutions.
The paper proves a Liouville theorem for special Lagrangian equations with convexity conditions.
problem Proving Liouville theorems for special Lagrangian equations with specific conditions.
method Using Neumann-Poincaré inequality, mean value inequality for superharmonic functions, and geometric measure theory.
result Derives global and interior Hessian estimates for solutions of special Lagrangian equations.
We prove the existence of non-smooth solutions to Special Lagrangian Equations in the non-convex case.
Study on radial solutions in higher dimensions, finding special cases.
problem Analyzing radial solutions to Hamiltonian stationary equations in various dimensions.
method Examined smooth radial solutions defined away from the origin, focusing on dimensions two and higher.
result In higher dimensions, non-special Lagrangian radial solutions exist near the origin, with continuity conditions.
We define the notion of special Lagrangian curvature, showing how it may be interpreted as an alternative higher dimensional generalisation of two dimensional Gaussian curvature. We obtain first a local rigidity result for this curvature when the ambiant manifold has negative sectional curvature. We then show how this …
We show that any global solution to the special Lagrangian equations with the phase larger than a critical value must be quadratic.
We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.
Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.
problem Solving special Lagrangian equations near infinity with specific conditions.
method Modified Kelvin transforms to characterize remainders in asymptotic expansions.
result Remainders in asymptotic expansions are characterized by a single smooth function in even dimensions and Cn−1,α in odd dimensions. We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.
We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.
We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three.
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.
Paper proves solvability condition for complex equation on special submanifolds.
problem Solvability condition for supercritical deformed Hermitian-Yang-Mills equation.
method Used integrals on subvarieties to provide necessary and sufficient condition.
result Confirms mirror version of Thomas-Yau conjecture about special Lagrangian submanifolds.
Quantization of a Lagrangian field system essentially depends on its degeneracy and implies its BRST extension defined by sets of non-trivial Noether and higher-stage Noether identities. However, one meets a problem how to select trivial and non-trivial higher-stage Noether identities. We show that, under certain condi…
Proves conjecture about special Lagrangians in G2-manifolds.
problem Existence of special Lagrangians in G2-manifolds.
method Solves real Monge-Ampère equation with singular right-hand side.
result Smoothness and asymptotic properties of special Lagrangians proved.
Construct special Lagrangian submanifolds in complex projective space.
problem Finding special Lagrangian submanifolds in complex projective space.
method Moment map technique and classification of cohomogeneity one actions.
result Examples of special Lagrangian submanifolds constructed.
The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
problem Mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
method Analyzes the parabolic equation and Monge-Ampère type equation, proving smooth solutions and convergence to self-expanding solutions.
result Smooth solutions u(x,t) for specific nonlinear equations and convergence to self-expanding solutions. Study proves existence of solutions for a specific type of parabolic equations.
problem Existence of solutions for second boundary value problem of parabolic equations.
method Established Schnextu¨rer's convergence result and applied it. result Existence of solutions for a family of special Lagrangian equations.
We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.
This is the first in a series of papers on special Lagrangian submanifolds in C^m. We study special Lagrangian submanifolds in C^m with large symmetry groups, and give a number of explicit constructions. Our main results concern special Lagrangian cones in C^m invariant under a subgroup G in SU(m) isomorphic to U(1)^{m…
Study of deformed Hermitian Yang-Mills equations with variable Kähler metrics.
problem Solving special Lagrangian type equations with variable metrics.
method Introducing extended gauge group to couple moment maps and scalar curvature.
result Solutions satisfy a mixture of K-stability and Bridgeland-type stability.
Identifies filtration in Lagrangian fibrations to monodromy weight filtration in degenerations.
problem Understanding the relationship between Lagrangian fibrations and degenerations of hyper-Kähler manifolds.
method Identifies and compares perverse filtration with monodromy weight filtration.
result Identifies the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a degeneration.
Develops methods to solve complex and real Hessian equations.
problem Solving complex and real Hessian equations on various domains.
method Introduces an ansatz to reduce PDEs to systems of ODEs, integrating via abelian integrals.
result Constructs entire solutions of arbitrary subcritical phase for dHYM/LYZ and special Lagrangian equations.
We study the Euler-Lagrange equations for a parameter dependent G-invariant Lagrangian on a homogeneous G-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group G, emphasizing the special invariance properties of the associated Euler-Poincaré equations with advected parameters.