Derives Hessian estimates for Lagrangian mean curvature equation.
problem Lagrangian mean curvature equation with supercritical phase and bounded second derivatives.
method Derives a priori interior Hessian estimates.
result Hessian estimates for Lagrangian mean curvature equation.
Solves a specific Dirichlet problem for Lagrangian mean curvature equations.
problem Solving the Dirichlet problem for Lagrangian mean curvature equations.
method Solves the Dirichlet problem for Lagrangian mean curvature equations on uniformly convex domains.
result Solves the Dirichlet problem for Lagrangian mean curvature equations.
Paper proves estimates for Lagrangian flow singularities.
problem Understanding Lagrangian flow singularities.
method Interior a priori estimates and Jacobi inequality.
result Proves estimates for supercritical Lagrangian phase.
Study shows unique tangent flow for Lagrangian surfaces with bounded mean curvature.
problem Understanding the behavior of Lagrangian surfaces with bounded mean curvature.
method Analyzing zero Maslov Lagrangian mean curvature flow in C2 with bounded mean curvature. result The tangent flow at a singular point is unique if the mean curvature stays uniformly bounded.
Study shows how to preserve Lagrangian condition in mean curvature flow on Kim-McCann metrics.
problem Preserving Lagrangian condition in mean curvature flow on Kim-McCann metrics.
method Expressed mean curvature flow within generalized mean curvature flow framework.
result Lagrangian condition is preserved along the flow.
The paper proves Hessian estimates for specific geometric flows.
problem Proving interior Hessian estimates for specific geometric flows.
method Proved interior Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian mean curvature flow.
result Extended results to a broader class of Lagrangian mean curvature type equations.
Paper proves gradient estimates for Lagrangian mean curvature equation.
problem Proving gradient estimates for Lagrangian mean curvature equation.
method Interior gradient estimates for critical and supercritical Lagrangian mean curvature equation.
result Solves Dirichlet boundary value problem for critical and supercritical Lagrangian mean curvature equation.
Ancient solutions and translators identified for Lagrangian flow.
problem Characterizing ancient solutions and translators of Lagrangian mean curvature flow.
method Analyzing almost calibrated, exact, ancient solutions with specific geometric properties.
result All ancient solutions with entropy less than 3 are special Lagrangian, planes, or translators in \(\mathbb{C}^2\).
Ancient Lagrangian flows get limited convex solutions.
problem Controlling convex solutions of Lagrangian flows at antiquity.
method Proving a Liouville type theorem with quadratic growth restrictions.
result Ancient convex solutions are unique.
The paper proves rigidity theorems for Type II singularities in Lagrangian flows.
problem Understanding Type II singularities in Lagrangian flows with zero Maslov class.
method Rigidity theorems for blow-up limits of Type II singularities.
result Generalized previous results from 2D to arbitrary dimensions.
The paper examines translating solitons and their relation to Lagrangian mean curvature flows with zero Maslov class.
problem Understanding the behavior of Lagrangian translating solitons near Type II singularities.
method Analyzes necessary conditions for blow-up limits and applies to open questions.
result Provides a necessary condition for blow-up limits of Lagrangian mean curvature flows with zero Maslov class.
In this paper, we derive a mean curvature estimate for eternal solutions (including translating solutions) of almost-calibrated Lagrangian mean curvature flow in complex Euclidean space. As a consequence, we show a non-existence result for eternal solutions of almost-calibrated Lagrangian mean curvature flow.
Study Hamiltonian stationary Lagrangian surfaces in complex space forms.
problem Characterize Lagrangian surfaces with harmonic mean curvature in complex space forms.
method Analyze surfaces with constant and harmonic mean curvature, using second fundamental form parallelism and Gaussian curvature constancy.
result Complete classification of Lagrangian surfaces with harmonic mean curvature and constant Gaussian curvature.
Convex solutions to a specific equation are smooth when the phase is smooth enough.
problem Regularity of solutions to the Lagrangian mean curvature equation.
method Showed regularity for convex solutions under Hölder continuity conditions on the phase.
result Convex viscosity solutions are regular if the Lagrangian phase is Hölder continuous.
In this paper, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclidean spaces. Moreover, our method can be applied to construct examples…
Given any embedded Lagrangian on a four dimensional compact Calabi-Yau, we find another Lagrangian in the same Hamiltonian isotopy class which develops a finite time singularity under mean curvature flow. This contradicts a weaker version of the Thomas-Yau conjecture regarding long time existence and convergence of Lag…
Proves flows of two-convex Lagrangians are regular, global, and converge.
problem Proves regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case.
method Uses a newly discovered monotone quantity to control two-convexity.
result Proves results for the mean curvature flow of area-decreasing Lagrangian submanifolds.
We introduce the notion of Kähler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a Kähler manifold that is almos…
Study on critical Lagrangian phase singularities in mean curvature flow.
problem Analyzing singularities in the Lagrangian mean curvature flow at the critical phase.
method Developed new method to prove C2,α estimates by using concave operators. result Established interior estimates for critical Lagrangian phase singularities.
Paper proves stability of flow in cotangent bundle for special Lagrangian submanifolds.
problem Stability of generalized Lagrangian mean curvature flow in cotangent bundle.
method New derivative estimates to weaken initial conditions and remove curvature constraints.
result Stability of flow near special Lagrangian submanifolds in cotangent bundle.
Study verifies Joyce's conjectures for circle-invariant Lagrangian surfaces.
problem Verifying Joyce's conjectures for specific Lagrangian surfaces.
method Continuation of Lagrangian mean curvature flow through finite time neck pinches.
result Flow converges to a chain of special Lagrangians, verifying conjectures.
The paper constructs singularities for Lagrangian flow in Gibbons-Hawking spaces with vanishing mean curvature.
problem Infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons-Hawking spaces.
method One-parameter family of barrier curves and detailed asymptotic analysis.
result The mean curvature converges uniformly to zero, but the second fundamental form becomes unbounded.
Two-dimensional Lagrangian mean curvature equation solved with new inequality.
problem Solving the two-dimensional Lagrangian mean curvature equation.
method Using Warren-Yuan's super isoperimetric inequality and simplified approach.
result Derivation of a modified Hessian bound for solutions.
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…
Classifies regularity for Lagrangian mean curvature type equations.
problem Classifying regularity for Lagrangian mean curvature type equations.
method Generalized constant rank theorem for Legendre transform, constructed convex solutions, and showed regularity conditions.
result Optimal regularity conditions for Lagrangian mean curvature type equations.
Study on singularities in Lagrangian mean curvature flow with special Lagrangian cones.
problem Understanding singularities in Lagrangian mean curvature flow.
method Analysis of tangent flows and blowup limits of special Lagrangian cones.
result Uniqueness of tangent flows in dimension two and any dimension when the link is connected.
Minimal Lagrangians in certain curved spaces are stable under specific flows.
problem Stability of minimal Lagrangians in Kähler-Einstein manifolds of non-positive curvature.
method Proved stability under Lagrangian mean curvature flow.
result Equivalence between linear and dynamical stability for C1-close Lagrangians. Constructs finite-time singularities in Lagrangian mean curvature flow with precise dynamics.
problem Finite-time singularities in Lagrangian mean curvature flow.
method Modulation analysis around shrinking cohomogeneity-one special Lagrangian desingularizations.
result Explicit curvature blow-up rate and precise dynamics of singularities.
It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.
We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-Hölder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian…
The abstract discusses special Lagrangians and their flow, proving conjectures and observing related phenomena.
problem Existence and long-time existence of special Lagrangian representatives and Lagrangian mean curvature flow.
method Gibbons-Hawking ansatz, circle-invariant hyperkaehler 4-manifolds, Calabi-Yau 2-folds, Thomas conjecture, Thomas-Yau conjecture.
result Proves versions of the Thomas conjecture and Thomas-Yau conjecture.
We prove rigidity of any properly immersed noncompact Lagrangian shrinker with single valued Lagrangian angle for Lagrangian mean curvature flows. Our pointwise approach also provides an ele- mentary proof to the known rigidity results for graphical and almost graphical shrinkers of mean curvature flows.
We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regular…
Flow preserves Lagrangian condition in Calabi-Yau manifolds.
problem Preserving Lagrangian condition in Calabi-Yau manifolds with boundary.
method Introduced mixed Dirichlet-Neumann boundary condition for Lagrangian mean curvature flow.
result Proved preservation of Lagrangian condition under flow.
In this paper we study the singularities of the mean curvature flow from a symplectic surface or from a Lagrangian surface in a Kähler-Einstein surface. We prove that the blow-up flow Σs∞ at a singular point (X0,T0) of a symplectic mean curvature flow Σt or of a Lagrangian mean curvature flow Σt is …
The paper proves existence and behavior of Lagrangian tori in complex projective plane.
problem Existence and behavior of Lagrangian tori in complex projective plane.
method Lagrangian mean curvature flow with surgery.
result Existence of monotone Lagrangian tori under Lagrangian mean curvature flow in complex projective plane.
Study cohomogeneity-one Lagrangian mean curvature flow in complex spaces.
problem Analyzing mean curvature flow of Lagrangians in complex spaces with specific group actions.
method Classifying solutions and singularities of cohomogeneity-one Lagrangian mean curvature flow.
result Explicit examples of new solitons and singularity models.
Proves better rigidity theorems for special solitons.
problem Understanding rigidity properties of specific solitons.
method Refined point-wise estimates for mean curvature.
result Stronger rigidity results for Lagrangian and symplectic translating solitons.
Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
problem Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
method Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
result Identifying the lowest eigenfunction with the scaling mode of the special Lagrangian desingularization
Study shows how neck pinches occur in Lagrangian flows and their continuation.
problem Understanding and continuation of Lagrangian mean curvature flows with singularities.
method Analyzes zero Maslov, rational Lagrangian flows in compact Calabi-Yau surfaces.
result Tangent flow is unique and can be continued past singularities.
This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold.
We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in Cn.
Inspired by the work of Leung-Wan, we study the mean curvature flow in hyperkähler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold, we define a new energy concept called …
In this note we prove a simple relation between the mean curvature form, symplectic area, and the Maslov class of a Lagrangian immersion in a Kähler-Einstein manifold. An immediate consequence is that in Kähler-Einstein manifolds with positive scalar curvature, minimal Lagrangian immersions are monotone.
The paper examines conditions for Lagrangian surfaces in Kähler-Einstein manifolds.
problem Characterizing Hamiltonian stationary Lagrangian surfaces with non-negative Gaussian curvature.
method Simple conditions and characterization of surfaces in Kähler-Einstein manifolds.
result Conditions for surfaces to have Euclidean factors or be fiber bundles over circles.
In this paper we investigate the singularities of Lagrangian mean curvature flows in Cm by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asympto…
In this paper, we study the generalized Lagrangian mean curvature flow in almost Einstein manifold proposed by T. Behrndt. We show that the singularity of this flow is characterized by the second fundamental form. We also show that the rescaled flow at a singularity converges to a finite union of Special Lagrangian con…
We survey some of the state of the art regarding singularities in Lagrangian mean curvature flow. Some open problems are suggested at the end.