A convergence result of the Lagrangian mean curvature flow

特殊拉格朗日曲率方程的先验内部估计

A priori interior estimates for special Lagrangian curvature equations

本文针对特殊拉格朗日曲率方程，在临界相和凸情形下建立了先验内部曲率估计，并证明了任意常数相的先验内部梯度估计。

古代拉格朗日平均曲率流的刘维尔型定理

A Liouville type theorem for ancient Lagrangian mean curvature flows

该论文证明了在受限二次增长假设下，古代拉格朗日平均曲率流的凸解满足刘维尔型定理，即解在特定条件下为常数或线性函数。

超临界相拉格朗日平均曲率流奇点的先验估计

A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase

本文针对超临界相拉格朗日平均曲率流，证明奇点的内部先验估计，建立临界及超临界下成立的雅可比不等式，并推广至更广的拉格朗日平均曲率型方程。

Gradient estimates for the Lagrangian mean curvature equation with critical and supercritical phase

In this paper, we prove interior gradient estimates for the Lagrangian mean curvature equation, if the Lagrangian phase is critical and supercritical and $C^{2}$. Combined with the a priori interior Hessian estimates proved in [Bha21, Bha22], this solves the Dirichlet boundary value problem for the critical and supercritical Lagrangian mean curvature equation with $C^0$ boundary data. We also provide a uniform gradient estimate for lower regularity phases that satisfy certain additional hypotheses.

On the entire self-shrinking solutions to Lagrangian mean curvature flow II

We show Bernstein type results for the entire self-shrinking solutions to Lagrangian mean curvature flow in $(\mathbb{R}^n\times\mathbb{R}^n, g_τ)$. The proofs rely on a priori estimates and barriers construction.

Convergence of Lagrangian mean curvature flow in Kähler-Einstein manifolds

In this paper, we give some convergence results of Lagrangian mean curvature flow under some stability conditions in a general Kähler-Einstein manifold. In particular, we prove that the flow will converge if the initial data is some small perturbation of stable minimal Lagrangian submanifold in a Kähler-Einstein manifold.

Lagrangian平均曲率方程的Dirichlet问题

The Dirichlet problem for Lagrangian mean curvature equation

本文在$\mathbb{R}^n$中一致凸的有界域上，求解了带连续边界数据的Lagrangian平均曲率方程的Dirichlet问题。

On the second boundary value problem for special Lagrangian curvature potential equation

This is a sequel to [1] and [2], which study the second boundary problem for special Lagrangian curvature potential equation. As consequences, we obtain the existence and uniqueness of the smooth uniformly convex solution by the method of continuity through carrying out a-priori estimate on the solutions.

Translating solutions to Lagrangian mean curvature flow

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal.

Some recent developments in Lagrangian mean curvature flows

We review some recent results on the mean curvature flows of Lagrangian submanifolds from the perspective of geometric partial differential equations. These include global existence and convergence results, characterizations of first-time singularities, and constructions of self-similar solutions.

The second type singularity of symplectic and Lagrangian mean curvature flows

In this paper we mainly study the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K ähler-Einstein surface. We show the relation between the maximum of the Kähler angle and the maximum of $|H|^2$ on the limit flow.

On short time existence of Lagrangian mean curvature flow

We consider a short time existence problem motivated by a conjecture of Joyce. Specifically we prove that given any compact Lagrangian $L\subset \mathbb{C}^n$ with a finite number of singularities, each asymptotic to a pair of non-area-minimising, transversally intersecting Lagrangian planes, there is a smooth Lagrangian mean curvature flow existing for some positive time, that attains $L$ as $t \searrow 0$ as varifolds, and smoothly locally away from the singularities.

On the second boundary value problem for Lagrangian mean curvature flow

We consider a fully nonlinear parabolic equation with nonlinear Neumann type boundary condition, and show that the longtime existence and convergence of the flow. Finally we apply this study to the boundary value problem for minimal Lagrangian graphs.

Mean curvature flow of Lagrangian submanifolds with isolated conical singularities

In this paper we study the short time existence problem for the (generalized) Lagrangian mean curvature flow in (almost) Calabi--Yau manifolds when the initial Lagrangian submanifold has isolated conical singularities modelled on stable special Lagrangian cones. Given a Lagrangian submanifold $F_0:L\rightarrow M$ in an almost Calabi--Yau manifold $M$ with isolated conical singularities at $x_1,...,x_n\in M$ modelled on stable special Lagrangian cones $C_1,...,C_n$ in $\mathbb{C}^m$, we show that for a short time there exist one-parameter families of points $x_1(t),... x_n(t)\in M$ and a one pa…

临界拉格朗日相位下拉格朗日平均曲率流的奇点

Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase

本文针对临界拉格朗日相位下的拉格朗日平均曲率流奇点建立内部估计，推广至更广方程类，并构造粘性解证明临界性必要性，同时引入新方法获得 $C^{2,\alpha}$ 估计。

余切丛中广义拉格朗日平均曲率流的稳定性

Stability of the generalized Lagrangian mean curvature flow in cotangent bundle

本文研究余切丛中图情形的广义拉格朗日平均曲率流稳定性，通过新的导数估计弱化初始条件并去除正曲率条件，证明特殊拉格朗日子流形附近该流是稳定的。

Lagrangian Mean Curvature flow for entire Lipschitz graphs II

We prove longtime existence and estimates for solutions to a fully nonlinear Lagrangian parabolic equation with locally $C^{1,1}$ initial data $u_0$ satisfying either (1) $-(1+η) I_n\leq D^2u_0 \leq (1+η)I_n$ for some positive dimensional constant $η$, (2) $u_0$ is weakly convex everywhere or (3) $u_0$ satisfies a large supercritical Lagrangian phase condition.

最优传输与通用拉格朗日平均曲率流在Kim-McCann度量上

Optimal Transport and Generalized Lagrangian Mean Curvature Flows on Kim-McCann Metrics

本文研究了伪黎曼流形上拉格朗日子流形的平均曲率流，揭示了Kim-McCann度量下para-Kähler几何技术的应用，证明了拉格朗日条件在流中的保持，并展示了Ma-Trudinger-Wang正则性理论在该设定下的适用性。

由Tonelli拉格朗日量诱导的最优输运问题的广义曲率

Generalized curvature for the optimal transport problem induced by a Tonelli Lagrangian

本文针对Tonelli拉格朗日量诱导的最优输运问题提出广义曲率概念，并证明其非负性可推出广义熵泛函在L-Wasserstein空间上沿C²位移插值的位移凸性。

Compactification of the space of Hamiltonian stationary Lagrangian submanifolds with bounded total extrinsic curvature and volume

For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^{k}$ for any nonnegative integer $k$ away from a finite set of points, and the limit is Hamiltonian stationary in ${\mathbb{C}}^{n}$. We also obtain a theorem on extending Hamiltonian stationary Lagrangian submanifolds $L$ across a compact set $N$ of Hausdorff codimension at least …

Recent Progress on Singularities of Lagrangian Mean Curvature Flow

We survey some of the state of the art regarding singularities in Lagrangian mean curvature flow. Some open problems are suggested at the end.

Finite Time Singularities for Lagrangian Mean Curvature 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 Lagrangian mean curvature flow.

Construction of Hamiltonian-stationary Lagrangian submanifolds of constant curvature $\varepsilon$ in complex space forms $\tilde M^n(4\varepsilon)

Lagrangian submanifolds of a Kaehler manifold are called Hamiltonian-stationary (or $H$-stationary for short) if it is a critical point of the area functional restricted to compactly supported Hamiltonian variations. In [B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Lagrangian isometric immersions of a real-space-form $M^{n}(c)$ into a complex-space-form $\tilde{M}^{n}(4c)$, Math. Proc. Cambridge Philo. Soc. 124 (1998), 107-125], an effective method to constructing Lagrangian submanifolds of constant curvature $\varepsilon$ in complex space form $M^n(4\varepsilon)$ was introduced. In …

A pointwise approach to rigidity of almost graphical self-shrinking solutions of mean curvature flows

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.

具有定量精确动力学的拉格朗日平均曲率流的有限时间奇点

Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics

构建几乎校准的拉格朗日平均曲率流，从接近特殊拉格朗日的初值出发，在有限时间产生II型奇点，给出曲率爆破率，奇点切流为特殊拉格朗日锥，爆破极限是光滑特殊拉格朗日去奇异化。

Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz

The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkaehler 4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian isotopy classes of Lagrangians, and the Thomas-Yau conjecture on long-time existence of the Lagrangian mean curvature flow. We also make observations concerning closed geodesics, curve shortening flow and minimal surfaces.

Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature

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.

Lagrange平均曲率型方程的最优正则性

Optimal regularity for Lagrangian mean curvature type equations

本文研究Lagrange平均曲率型方程凸解的正则性，分别在$C^2$凸相位和高Hölder指数条件下证明正则性，并用奇异解证明条件与指数最优。

A note on the two dimensional Lagrangian mean curvature equation

In this note, we use Warren-Yuan's super isoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two dimensional Lagrangian mean curvature equation. We assume the Lagrangian phase to be supercritical with bounded second derivatives. Unlike the previous approach, the simplified approach in this proof does not require the Michael-Simon mean value and Sobolev inequalities on generalized submanifolds of $\mathbb{R}^n$.

Lagrangian angles of family of Lagrangian fibrations under mean curvature flow

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 under mean curvature flow.

Singularities of Equivariant Lagrangian Mean Curvature Flow

We study almost-calibrated, $O(n)$-equivariant Lagrangian mean curvature flow in $\mathbb{C}^n$, and prove structural theorems about the Type I and Type II blowups of finite-time singularities. In particular, we prove that any Type I blowup of such a flow must be a special Lagrangian pair of transversely intersecting planes, any Type II blowup must be the Lawlor neck with the same asymptotes, and these blowups are independent of the choice of rescaling. We also give a partial classification of when singularities occur in the equivariant case, and examine the intermediate scales between the Typ…

具有Lipschitz临界和超临界相位的Lagrangian平均曲率方程的Hessian估计

Hessian estimates for Lagrangian mean curvature equation with Lipschitz critical and supercritical phases

本文提出新策略，结合Allard型正则性定理与几何测度论等方法，在锐利的Lipschitz临界和超临界相位条件下，获得了Lagrangian平均曲率方程解的内部Hessian估计。

平均曲率在拉格朗日平均曲率流的非退化颈缩奇点处爆破

The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow

本文证明拉格朗日平均曲率流在非退化颈缩奇点处平均曲率无界，并给出爆破速率控制。

Weighted Hamiltonian stationary Lagrangian submanifolds and generalized Lagrangian mean curvature flows in toric almost Calabi-Yau manifolds

In this paper we generalize examples of Hamiltonian stationary Lagrangian submanifolds constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau manifolds. We construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized Lagrangian mean curvature flows starting from these examples. We allow these flows to have some singularities and topological changes.

Lagrangian平均曲率流的短时间存在性

Short-time existence of Lagrangian mean curvature flow

本文利用直接的P.D.E.方法和带corners的流形概念，证明了Lagrangian平均曲率流在更强意义下收敛的短时间存在性解。

Some Inequalities Related to Ricci Curvatures for Lagrangian Submanifolds of Kahler QCH-manifolds

By establishing two general quadratic inequalities, we obtain some inequalities related to Ricci curvatures for Lagrangian submanifolds of K$\ddot{\mathrm{a}}$hler QCH-manifolds, which generalize some results for Lagrangian submanifolds of complex space forms.

On the Gluing Construction of Translating Solitons to Lagrangian Mean Curvature Flow

We construct Lagrangian translating solitons by desingularizing the intersection points between Lagrangian Grim Reaper cylinders with the same phase using special Lagrangian Lawlor necks. The resulting Lagrangian translating solitons could have arbitrarily many ends and non-contractible loops.

Mean curvature flow of monotone Lagrangian submanifolds

We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in $\mathbb C^{n}$.

The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow

We provide several rigidity results for the Clifford torus in the class of compact self-shrinkers for Lagrangian mean curvature flow.

具有平均曲率界的曲面的拉格朗日平均曲率流

Lagrangian mean curvature flow of surfaces with mean curvature bound

本文证明在平均曲率一致有界条件下，零 Maslov 拉格朗日平均曲率流在奇异点处的切流唯一，即抛物重标度极限与重标度序列无关。

Hamiltonian stationary self-similar solutions for Lagrangian mean curvature flow in complex Euclidean plane

We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.

Local non-collapsing of volume for the Lagrangian mean curvature flow

We prove an optimal control on the time-dependent measure of a measurable set under a reparametrized Lagrangian mean curvature flow of almost calibrated submanifolds in a Calabi-Yau manifold. Moreover we give a classification of those Lagrangian translating solitons in $\mathbb{C}^m$ that evolve by this reparametrized flow

零Maslov类的Lagrangian平均曲率流的Type II奇点

Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class

本文证明了零Maslov类或几乎校准的Lagrangian平均曲率流在Type II奇点处爆破极限的刚性定理，将二维结果推广到任意维数。

Singularities of symplectic and Lagrangian mean curvature flows

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 $Σ_s^\infty$ at a singular point $(X_0, T_0)$ of a symplectic mean curvature flow $Σ_t$ or of a Lagrangian mean curvature flow $Σ_t$ is a non trivial minimal surface in ${\bf R}^4$, if $Σ_{-\infty}^\infty$ is connected.

Examples of equivariant Lagrangian mean curvature flow

In this expository note we describe important examples of Lagrangian mean curvature flow in $\mathbb{C}^2$ which are invariant under a circle action. Through these examples, we see compact and non-compact situations, long-time existence, singularities forming via explicit models, and significant objects in Riemannian and symplectic geometry, including the Clifford torus, Chekanov torus, Whitney sphere and Lawlor necks.

Generalized Lagrangian mean curvature flow in Kähler manifolds that are almost Einstein

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 almost Einstein, and which in addition has a trivial canonical bundle, we show that the generalized mean curvature vector field of a Lagrangian submanifold is the dual vector field associated to the Lagrangian angle.

Singularities of the Lagrangian Mean Curvature Flow

In this paper we investigate the singularities of Lagrangian mean curvature flows in $\mathbf{C}^m$ 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 asymptotic to special Lagrangian cones; hence all type II singularities are modeled by unions of special Lagrangian cones.

具有圆对称性的拉格朗日平均曲率流中的颈缩奇点与Joyce猜想

Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry

本文研究具有圆对称性的hyperkähler 4-流形中的拉格朗日平均曲率流，证明其可穿过有限时间奇点延续并收敛至special Lagrangians链，验证了Joyce猜想并指出奇点为颈缩。

具有谐波平均曲率的Hamiltonian stationary Lagrangian曲面

Hamiltonian stationary Lagrangian surfaces with harmonic mean curvature in complex space forms

本文研究复空间形式中具有谐波平均曲率的Hamiltonian stationary Lagrangian曲面，通过分析第二基本形式和高斯曲率的条件，实现了此类曲面的完整分类。

余一性拉格朗日平均曲率流

Cohomogeneity-One Lagrangian Mean Curvature Flow

本文研究余一性拉格朗日平均曲率流，分类自相似解与特殊拉格朗日子流形，刻画奇点爆破模型，并给出无穷多新孤子与奇点模型例子。

拉格朗日平均曲率方程解的非唯一性

Nonuniqueness of solutions to the Lagrangian mean curvature equation

该研究解决了Harvey和Lawson的问题，证明对于n≥2的所有维度，单位球上带连续边界数据的拉格朗日平均曲率方程狄利克雷问题存在连续的不同粘性解的连续统，即解不唯一。

Gibbons--Hawking空间中具有消失平均曲率的拉格朗日平均曲率流的无限时间奇点

Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons--Hawking Spaces

本文在Gibbons-Hawking空间中构造了拉格朗日平均曲率流的无限时间奇点，其中平均曲率趋于零但第二基本形式无界，并证明了曲率爆破速率为log max|A| ~ sqrt(t)。

The Mean Curvature of Special Lagrangian 3-folds in SU(3)-Structures with Torsion

We derive formulas for the mean curvature of special Lagrangian 3-folds in the general case where the ambient 6-manifold has intrinsic torsion. Consequently, we are able to characterize those SU(3)-structures for which every special Lagrangian 3-fold is a minimal submanifold. In the process, we obtain an obstruction to the local existence of special Lagrangian 3-folds.

Generalized Lagrangian mean curvature flows in almost Calabi-Yau manifolds

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 cones for generalized Lagrangian mean curvature flow with zero-Maslov class in almost Calabi-Yau manifold. As a corollary, there is no finite time Type-I singularity for such a flow.