Tight Lagrangian surfaces in $S^2 \times S^2$

Lagrangian Bonnet pairs in complex space forms

In this paper we first give a Bonnet theorem for conformal Lagrangian surfaces in complex space forms, then we show that any compact Lagrangian surface in the complex space form admits at most one other global isometric Lagrangian surface with the same mean curvature form, unless the Maslov form is conformal. These two Lagrangian surfaces are then called Lagrangian Bonnet pairs. We also studied the question about Lagrangian Bonnet surfaces in $\tilde{M}^2(4c)$, and obtain some interesting results.

Evolution of spacelike surfaces in anti-De Sitter space by their Lagrangian angle

We study spacelike hypersurfaces in anti-De Sitter spacetime that evolve by the Lagrangian angle of their Gauß maps.

A new look at equivariant minimal Lagrangian surfaces in $\mathbb{C}P^2$

In this note, we present a new look at translationally equivariant minimal Lagrangian surfaces in the complex projective plane via the loop group method.

$"H^2 \ imes H^2$ 中的拉格朗日曲面

Lagrangian surfaces in $\mathbb H^2 \times \mathbb H^2$

本文研究双曲平面乘积空间中的拉格朗日曲面，分类了全脐、全测地、平行第二基本形式、常高斯曲率极小及满足有界条件的完备极小拉格朗日曲面，并给出若干例子。

A note on Lagrangian submanifolds of twistor spaces and their relation to superminimal surfaces

In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian $4$-manifold and particular Lagrangian submanifolds of the twistor space over the $4$-manifold is proven. More explicitly, for every superminimal surface a submanifold of the twistor space is constructed which is Lagrangian for all the natural almost Hermitian structures on the twistor space. The twistor fibration restricted to the constructed Lagrangian gives a circle bundle over the superminimal surface. Conversely, if a submanifold of the twistor space is Lagrangian for all the natural almost He…

Ruh-Vilms Theorems For Minimal Surfaces Without Complex Points and Minimal Lagrangian Surfaces in $\mathbb C P^2$

In this paper we investigate surfaces in $\mathbb C P^2$ without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh-Vilms type theorems. We also discuss the liftability of an immersion from a surface to $\mathbb C P^2$ into $S^5$ in Appendix A.

Symplectic Applicability of Lagrangian Surfaces

We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit examples are considered.

Rigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary

In this note, we study minimal Lagrangian surfaces in $\mathbb{B}^4$ with Legendrian capillary boundary on $\mathbb{S}^3$. On the one hand, we prove that any minimal Lagrangian surface in $\mathbb{B}^4$ with Legendrian free boundary on $\mathbb{S}^3$ must be an equatorial plane disk. One the other hand, we show that any annulus type minimal Lagrangian surface in $\mathbb{B}^4$ with Legendrian capillary boundary on $\mathbb{S}^3$ must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng (Sci. China Math., 2020).

On Hamiltonian stationary Lagrangian spheres in non-Einstein Kaehler surfaces

Hamiltonian stationary Lagrangian spheres in Kaehler-Einstein surfaces are minimal. We prove that in the family of non-Einstein Kaehler surfaces given by the product $Σ_1\timesΣ_2$ of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example is defined when the surfaces $Σ_1$ and $ Σ_2$ are spheres.

Null pseudo-isotropic Lagrangian surfaces

In this paper we will show that a Lagrangian, Lorentzian surface $M^2_1$ in a complex pseudo space form $\widetilde M^2_1 (4c)$ is pseudo-isotropic if and only if $M$ is minimal. Next we will obtain a complete classification of all Lagrangian, Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isotropic.

Hamiltonian stationary Lagrangian surfaces in C^2

We study Hamiltonian stationary Lagrangian surfaces in C^2, i.e. Lagrangian surfaces in C^2 which are stationary points of the area functional under smooth Hamiltonian variations. Using loop groups, we propose a formulation of the equation as a completely integrable system. We construct a Weierstrass type representation and produce all tori through either the integrable systems machinery or more direct arguments.

Smoothing singular extremal Kähler surfaces and minimal Lagrangians

We consider smoothings of a complex surface with singularities of class T and no nontrivial holomorphic vector field. Under an hypothesis of non degeneracy of the smoothing at each singular point, we prove that if the singular surface admits an extremal metric, then the smoothings also admit extremal metrics in nearby Kähler classes. In addition, we construct small Lagrangian stationary spheres which represent Lagrangian vanishing cycles for surfaces close to the singular one.

Explicit expressions for the Iwasawa factors, the metric and the monodromy matrices for minimal Lagrangian surfaces in $\mathbb{C} P^2$

In this paper we continue our study of equivariant minimal Lagrangian surfaces in $\mathbb{C} P^2$, characterizing the rotationally equivariant cases and providing explicit formulae for relevant geometric quantities of translationally equivariant minimal Lagrangian surfaces in terms of Weierstrass elliptic functions.

Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space

It has been known for some time that there exist $5$ essentially different real forms of the complex affine Kac-Moody algebra of type $A_2^{(2)}$ and that one can associate $4$ of these real forms with certain classes of "integrable surfaces", such as minimal Lagrangian surfaces in $\mathbb {CP}^2$ and $\mathbb {CH}^2$, as well as definite and indefinite affine spheres in $\mathbb R^3$. In this paper we consider the class of timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space $\mathbb{CH}^2_1$. We show that this class of surfaces corresponds to the fifth real fo…

Kähler-Einstein曲面中具有非负高斯曲率的哈密顿平稳拉格朗日曲面

Hamiltonian Stationary Lagrangian Surfaces with Non-Negative Gaussian Curvature in Kähler-Einstein Surfaces

该研究在微分几何领域，给出Kähler-Einstein流形中哈密顿平稳拉格朗日子流形的结构判定条件，并刻画了单连通复空间形式中满足非负高斯曲率、平均曲率向量在$L^p$空间的哈密顿平稳拉格朗日曲面。

Quasi-biharmonic Lagrangian surfaces in Lorentzian complex space forms

In this paper, we introduce the notion of a quasi-biharmonic submanifold in a pseudo-Riemannian manifold and classify quasi-biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms.

Cyclic and ruled Lagrangian surfaces in complex Euclidean space

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve of the 3-sphere or a Legendrian curve of the anti de Sitter 3-space. We also describe ruled Lagrangian surfaces. Finally we characterize those cyclic and ruled Lagrangian surfaces which are solutions to the self-similar equation of the Mean Curvature Flow. Finally, we give a partial result in the case of Hamiltonian stationary cyclic surfaces…

Minimizing area among Lagrangian surfaces: the mapping problem

This paper introduces a geometrically constrained variational problem for the area functional. We consider the area restricted to the langrangian surfaces of a Kaehler surface, or, more generally, a symplectic 4-manifold with suitable metric, and study its critical points and in particular its minimizers. We apply this study to the problem of finding canonical representatives of the lagrangian homology (that part of the homology generated by lagrangian cycles).

通过环路群方法研究 $\mathbb{C}P^2$ 中的极小拉格朗日曲面 第二部分：一般情形

Minimal Lagrangian surfaces in $\mathbb{C}P^2$ via the loop group method Part II: The general case

本文将环路群方法从可缩黎曼曲面推广到任意黎曼曲面，构造 $\mathbb{C}P^2$ 中的极小拉格朗日浸入与柱面，并证明其逼近 Delaunay 柱面。

Minimal Lagrangian surfaces in $S^2 \times S^2$

We deal with the minimal Lagrangian surfaces of the Einstein-Kähler surface $S^2 \times S^2$, studying local geometric properties and showing that they can be locally described as Gauss maps of minimal surfaces in $S^3 \subset R^4$. We also discuss the second variation of the area and characterize the most relevant examples by their stability behaviour.

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。

Lagrangian submanifolds of the nearly Kähler $\mathbb{S}^3 \times \mathbb{S}^3$ from minimal surfaces in $\mathbb{S}^3$

We study non-totally geodesic Lagrangian submanifolds of the nearly Kähler $\mathbb{S}^3 \times \mathbb{S}^3$ for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in $\mathbb{S}^3$. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example.

On the Lagrangian angle and the Kähler angle of immersed surfaces in the complex plane $\bbc^2$

In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in $\mathbb C^2$. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in $\mathbb C^2$ than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant Kähler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant Kähler angle is generally non-vanishing. Secondly, we obtain two pinching results for the Kähler angle which imply rigidity theorems of self-shr…

二维复空间形式中具有常高斯曲率的双调和拉格朗日曲面

Biharmonic Lagrangian surfaces with constant Gaussian curvature in $2$-dimensional complex space forms

本文在二维复空间形式中完整分类了具有常高斯曲率的双调和拉格朗日曲面，并由此分类了平坦法丛情形，证明了伪脐双调和拉格朗日曲面必为极小曲面。

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Given an oriented Riemannian surface $(Σ, g)$, its tangent bundle $TΣ$ enjoys a natural pseudo-Kähler structure, that is the combination of a complex structure $\J$, a pseudo-metric $\G$ with neutral signature and a symplectic structure $\Om$. We give a local classification of those surfaces of $TΣ$ which are both Lagrangian with respect to $\Om$ and minimal with respect to $\G$. We first show that if $g$ is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly…

Formal Killing fields for minimal Lagrangian surfaces in complex space forms

The differential system for minimal Lagrangian surfaces in a $2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic system defined on the bundle of oriented Lagrangian planes. This is a 6-symmetric space associated with the Lie group SL(3,$\mathbb{C}$), and the minimal Lagrangian surfaces arise as the primitive maps. Utilizing this property, we derive the differential algebraic inductive formulas for a pair of loop algebra $\mathfrak{sl}(3,\mathbb{C})[[λ]]$-valued canonical formal Killing fields. As a result, we give a complete classification of the (infinite sequence of) Ja…

复空间形式中平均曲率梯度长度恒定的哈密顿稳态拉格朗日曲面

Hamiltonian stationary Lagrangian surfaces in complex space forms with constant-length gradient of the mean curvature

本文证明在复空间形式中，若哈密顿稳态拉格朗日曲面的高斯曲率恒定且平均曲率梯度长度为正常数，则曲面与空间必平坦，且局部为柯努螺线乘积曲面。

具有勒让德边界的拉格朗日曲面

Lagrangian surfaces with Legendrian boundary

本文引入拉格朗日子流形的边界问题，给出满足条件的例子，并证明尼切型定理，提出相关问题。

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.

Complex and Lagrangian surfaces of the complex projective plane via Kählerian Killing Spin$^c$ spinors

The complex projective space $\mathbb C P^2$ of complex dimension $2$ has a Spin$^c$ structure carrying Kählerian Killing spinors. The restriction of one of these Kählerian Killing spinors to a surface $M^2$ characterizes the isometric immersion of $M^2$ into $\mathbb C P^2$ if the immersion is either Lagrangian or complex.

Lagrangian surfaces with circullar ellipse of curvature in complex space forms

We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus of the complex projective plane and of the Whitney spheres in the complex projective, complex Euclidean and complex hyperbolic planes.

通过环群方法研究二维复双曲二次曲面的极小拉格朗日面

Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method

本文研究复双曲二次曲面中的极小拉格朗日面，通过环群方法证明极小性由平坦连接的环表示特征化，并建立与反德西特3空间中类空最大面的对应关系，进而发展出DPW型表示并构造出具体的例子，包括R-等价和径向对称面。

Neck pinches along the Lagrangian mean curvature flow of surfaces

Let $L_t$ be a zero Maslov, rational Lagrangian mean curvature flow in a compact Calabi-Yau surface, and suppose that at the first singular time a tangent flow is given by the static union of two transverse planes. We show that in this case the tangent flow is unique, and that the flow can be continued past the singularity as an immersed, smooth, zero Maslov, rational Lagrangian mean curvature flow. Furthermore, if $L_0$ is a sphere that is stable in the sense of Thomas-Yau, then such a singularity cannot form.

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。

Lagrangian surfaces in complex Euclidean plane via spherical and hyperbolic curves

We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hamiltonian-minimal and Willmore surfaces in terms of simple properties of the curvature of the generating curves. As applications, we provide explicitly conformal parametrizations of known and new examples of minimal, constant mean curvature, Hamiltonian-minimal and Willmore surface…

Evolution by mean curvature flow of Lagrangian spherical surfaces in complex Euclidean plane

We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane $\mathbb{C}^2$. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in $\mathbb{C}^2$ such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the Clifford torus.

Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric spaces

This paper is the third of a series on Hamiltonian stationary Lagrangian surfaces. We present here the most general theory, valid for any Hermitian symmetric target space. Using well-chosen moving frame formalism, we show that the equations are equivalent to an integrable system, generalizing the C^2 subcase analyzed in the first article (arXiv:math.DG/0009202). This system shares many features with the harmonic map equation of surfaces into symmetric spaces, allowing us to develop a theory close to Dorfmeister, Pedit and Wu's, including for instance a Weierstrass-type representation. Notice t…

复空间形式中具有常高斯曲率的哈密顿平稳拉格朗日曲面

Hamiltonian stationary Lagrangian surfaces of constant Gaussian curvature in complex space forms

针对复空间形式中具有常高斯曲率的哈密顿平稳拉格朗日曲面，研究其平均曲率向量场的性质，证明该向量场要么为常数，要么满足特定条件，且当平均曲率向量场非常数时存在对应结论。

Minimal surfaces in $\mathbb{R}^4$ like the Lagrangian catenoid

In this paper, we discuss complete minimal immersions in $\mathbb{R}^N$($N\geq4$) with finite total curvature and embedded planar ends. First, we prove nonexistence for the following cases: (1) genus 1 with 2 embedded planar ends, (2) genus $\neq4$, hyperelliptic with 2 embedded planar ends like the Lagrangian catenoid. Then we show the existence of embedded minimal spheres in $\mathbb{R}^4$ with 3 embedded planar ends. Moreover, we construct genus $g$ examples in $\mathbb{R}^4$ with $d$ embedded planar ends such that $g\geq 1$ and $g+2\leq d\leq 2g+1$. These examples include a family of embed…

Symmetry groups and Lagrangians associated to Tzitzeica surfaces

One applies the symmetry group theory for study the partial differential equations of Tzitzeica surfaces theory. One finds infinitesimal symmetries, Lagrangians and a new solution of Titzeica equation.

Minimal surface systems, maximal surface systems, and special Lagrangian equations

We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.

On a new construction of special Lagrangian immersions in complex Euclidean space

We construct new special Lagrangian submanifolds in complex Euclidean space using a pair of minimal Legendrian submanifolds in odd-dimensional spheres and certain Lagrangian surface belonging to a family that can be considered as a generalization of the special Lagrangian surfaces in complex Euclidean plane. Our examples include those invariant under the standard action of SO(p+1)xSO(q+1) on C^n = C^(p+1) x C^(q+1), n=p+q+2.

Lagrangian spheres in Del Pezzo surfaces

Lagrangian spheres in the symplectic Del Pezzo surfaces arising as blow-ups of the complex projective plane in 4 or fewer points are classified up to Lagrangian isotopy. Unlike the case of the 5-point blow-up, there is no Lagrangian knotting.

A Converging Lagrangian Curvature Flow in the Space of Oriented Lines

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the 3-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is determined by the initial surface. To prove this we show that the oriented normal lines, considered as a surface in the space of all oriented lines, evolve by a parabolic flow which preserves the Lagrangian condition. Moreover, this flow converges to a holomorphic Lagrangian section, which form the set of oriented lines t…

Explicit gradient estimates for minimal Lagrangian surfaces of dimension two

We derive explicit, uniform, a priori interior Hessian and gradient estimates for special Lagrangian equations of all phases in dimension two.

Singular Solutions to Special Lagrangian Equations with Subcritical Phases and Minimal Surface Systems

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.

Conformal geometry of surfaces in the Lagrangian--Grassmannian and second order PDE

Of all real Lagrangian--Grassmannians $LG(n,2n)$, only $LG(2,4)$ admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite M\"obius space $S^{1,2}$. Using Cartan's method of moving frames, we study hyperbolic (timelike) surfaces in $LG(2,4)$ modulo the conformal symplectic group $CSp(4,R)$. This $CSp(4,R)$-invariant classification is also a contact-invariant classification of (in general, highly non-linear) second order scalar hyperbolic PDE in the plane. Via $LG(2,4)$, we give a simple geometric argument for the invariance of the general hyperbolic M…

Lagrangian immersions in the product of Lorentzian two manifold

For Lorentzian 2-manifolds $(Σ_1,g_1)$ and $(Σ_2,g_2)$ we consider the two product para-Kähler structures $(G^ε,J,Ω^ε)$ defined on the product four manifold $Σ_1\timesΣ_2$, with $ε=\pm 1$. We show that the metric $G^ε$ is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures $κ_1,κ_2$ of $g_1,g_2$, respectively, are both constants satisfying $κ_1=-εκ_2$ (resp. $κ_1=εκ_2$). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel mean curvature vector is the product $γ_1\timesγ_2\subsetΣ_1\timesΣ_2$, where $γ_1$ and $γ_2$ are curves…

极小 $δ(2)$-理想拉格朗日子流形与四元数射影空间

Minimal $δ(2)$-ideal Lagrangian submanifolds and the Quaternionic projective space

本文构造显式映射将 $\mathbb{C}^n$ 的极小 $δ(2)$-理想拉格朗日子流形映至四元数射影空间，并在 $n=3$ 时建立其与极小全复曲面及极小拉格朗日曲面间的一一对应关系。

K3曲面上的特殊拉格朗日子流形与圆塌缩

Special Lagrangian submanifolds and circle collapse on K3

研究K3曲面塌缩到三维仿射基时，仿射线提升为退化特殊拉格朗日球面和环面的序列，并构造连接Taub-NUT气泡对的特殊拉格朗日球面。

On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Let $(Σ_1,g_1)$ and $(Σ_2,g_2)$ be connected, complete and orientable Riemannian two manifolds. Consider the two canonical Kähler structures $(G^ε,J,Ω^ε)$ on the product 4-manifold $Σ_1\timesΣ_2$ given by $ G^ε=g_1\oplus εg_2$, $ε=\pm 1$ and $J$ is the canonical product complex structure. Thus for $ε=1$ the Kähler metric $G^+$ is Riemannian while for $ε=-1$, $G^-$ is of neutral signature. We show that the metric $G^ε$ is locally conformally flat iff the Gauss curvatures $κ(g_1)$ and $κ(g_2)$ are both constants satisfying $κ(g_1)=-εκ(g_2)$. We also give conditions on the Gauss curvatures for wh…

通过环群方法研究二维复二次曲面的极小拉格朗日面

Minimal Lagrangian surfaces in the two dimensional complex quadric via the loop group method

本文通过环群方法研究复二次曲面中的极小拉格朗日面，建立了与$\mathbb S^{3}$中极小面的对应关系，并给出了多种显式例子。

Hamiltonian stability and index of minimal Lagrangian surfaces of the complex projective plane

We show that the Clifford torus and the totally geodesic real projective plane RP^2 in the complex projective plane CP^2 are the unique Hamiltonian stable minimal Lagrangian compact surfaces of CP^2 with genus less than or equal to 4, when the surface is orientable, and with Euler characteristic greater than or equal to -1, when the surface is nonorientable. Also we characterize RP^2 in CP^2 as the least possible index minimal Lagrangian compact nonorientable surface of CP^2.

The Deformation of Lagrangian Minimal Surfaces in Kahler-Einstein Surfaces

Let $(N,g_{0})$ be a Kahler-Einstein surface with the first Chern class negative and assume that there exists a branched Lagrangian minimal surfaces with respect to the metric $g_{0}$. We show that when the Kahler-Einstein metric is changed in the same component (i.e. the complex structure is changed), the Lagrangian minimal surface can be deformed accordingly. To get the result, we first obtain a theorem on the deformation of the branched minimal surfaces in a complete Riemannian $n$-manifold and also generalize a result of J. Chen and G. Tian on the limit of adjunction numbers.