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.
Paper shows no eternal solutions for certain flows.
problem Existence of eternal solutions for Lagrangian mean curvature flow.
method Derived mean curvature estimate for eternal solutions.
result Non-existence of eternal solutions for almost-calibrated Lagrangian mean curvature flow.
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 constructs examples of Lagrangian flows using moment maps.
problem Constructing Lagrangian mean curvature flows in Calabi-Yau manifolds.
method Using moment maps for abelian Lie group actions.
result Examples of Lagrangian self-shrinkers and translating solitons.
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.
Lagrangian spheres develop singularities under flow, matching Whitney spheres.
problem Understanding singularities in Lagrangian mean curvature flow.
method Analyzing equivariant Lagrangian spheres with Ricci curvature conditions.
result Whitney spheres develop type-II singularities rescaling to a grim reaper and flat subspace.
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 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.
Proves rigidity of certain Lagrangian shrinkers using a pointwise approach.
problem Rigidity of properly immersed noncompact Lagrangian shrinkers with single valued Lagrangian angle.
method Pointwise approach to prove rigidity of shrinkers.
result Elementary proof of known rigidity results for graphical and almost graphical shrinkers.
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.
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 \(\mathbb{C}^2\).
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.
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.
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 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.
Study of mean curvature flow in hyperkähler manifolds leading to complex Lagrangian submanifolds.
problem Understanding the behavior of mean curvature flow in hyperkähler manifolds.
method Definition of twistor energy and analysis of mean curvature flow starting from hyper-Lagrangian submanifolds.
result Mean curvature flow converges to a complex Lagrangian submanifold for sufficiently small twistor energy.
Study Bernstein results for self-shrinking solutions in Lagrangian flow.
problem Understanding entire self-shrinking solutions in Lagrangian flow.
method Prior estimates and barriers construction.
result Showed Bernstein type results for self-shrinking solutions.
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.
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…
Study controls volume measure for Lagrangian flows in Calabi-Yau manifolds.
problem Controlling volume measure for Lagrangian flows in Calabi-Yau manifolds.
method Optimal control on time-dependent measure of a measurable set under reparametrized Lagrangian mean curvature flow.
result Classification of Lagrangian translating solitons in Cm that evolve by the reparametrized flow. 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 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.
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…
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. 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.
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.
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 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.
Study shows Whitney sphere collapses to a point in finite time.
problem Understanding the evolution of Whitney sphere under mean curvature flow.
method Investigated equivariant Lagrangian spheres in \(\mathbb{C}^n\) using mean curvature flow.
result Equivariant Lagrangian spheres collapse to a point in finite time and converge to a plane with multiplicity two.
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.
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 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…
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…
Study of Lagrangian mean curvature flow with equivariant symmetry.
problem Understanding singularities in Lagrangian mean curvature flow.
method Structural theorems about blowups of finite-time singularities.
result Classification of singularities in equivariant case.
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.
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 establish the longtime existence and convergence results of the mean curvature flow of entire Lagrangian graphs in Pseudo-Euclidean space which is related to Logarithmic gradient flow.
Proves existence of Lagrangian mean curvature flow solutions.
problem Desingularizing transverse intersection points of immersed Lagrangians.
method Direct PDE approach using manifolds with corners and a-corners.
result Existence of Lagrangian mean curvature flow solutions with stronger convergence.
We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in Cn.
The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.
problem Infinite-time singularities in Lagrangian mean curvature flow.
method Constructing solutions by gluing special Lagrangian 'Lawlor necks' and analyzing dynamics of neck size.
result The flow decomposes initial data into a union of special Lagrangians intersecting at one point.
Ancient solutions in Lagrangian flow are classified based on their blow-down.
problem Understanding ancient solutions in Lagrangian mean curvature flow.
method Structural and classification results for ancient solutions, focusing on the almost calibrated case.
result Classification of Type II blow-ups in terms of their blow-down.
We survey some of the state of the art regarding singularities in Lagrangian mean curvature flow. Some open problems are suggested at the end.
This note surveys and compares results on the separation of variables construction for soliton solutions of curvature equations including the Kähler-Ricci flow and the Lagrangian mean curvature flow. In the last section, we propose some new generalizations in the Lagrangian mean curvature flow case.
We provide several rigidity results for the Clifford torus in the class of compact self-shrinkers for Lagrangian mean curvature flow.
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
We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.
Study on evolving graphs of functions under mean curvature flow in R^n.
problem Proving long-time existence and convergence of special Lagrangian evolution equation.
method Consider the graph of a C2 function u on Rn, deform it by mean curvature flow, and analyze under 2-positivity assumption. result Proves long-time existence and convergence results under 2-positivity assumption, improving previous results.