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

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.

$\nmathbb{R}$-等变极小拉格朗日曲面在复二次曲面 $Q_{2}$ 与复双曲二次曲面 $Q_{2}^{*}$ 中

$\mathbb{R}$-equivariant minimal Lagrangian surfaces in the complex quadric $Q_{2}$ and the complex hyperbolic quadric $Q_{2}^{*}$

研究复二次曲面与复双曲二次曲面中由环群势产生的等变极小拉格朗日曲面，给出闭合判据、稳定性分析及宽度条件。

Equivariant CR minimal immersions from $S^3$ into $\mathbb{C}P^n$

The equivariant CR minimal immersions from the round $3$-sphere $S^3$ into the complex projective space $\mathbb CP^n$ have been classified by the third author explicitly (J London Math Soc 68: 223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant, and that all geometric properties of $S^3={\rm SU}(2)$ endowed with a left-invariant metric can be expressed in terms of the structure constants of the Lie algebra $\mathfrak{su}(2)$, we establish an extended classification theorem for equivariant CR minimal immersions from the …

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。

等变极小超曲面的无限存在性

Infinite existence of equivariant minimal hypersurfaces

研究证明闭流形上存在无限多个等变极小超曲面，并建立等变极小最大理论。

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。

极小极大$G$-不变极小超曲面的等变Morse指标

Equivariant Morse index of min-max $G$-invariant minimal hypersurfaces

本文研究等变极小极大理论下的$G$-不变极小超曲面，证明其紧性定理并构造极限上的$G$-不变Jacobi场，得到通有有限性结果，还将Morse指标估计推广到等变情形，证明$k$维同伦类构造的超曲面等变指标不超过$k$。

等变艾伦 - 卡恩解与上同调维数为2的极小曲面的存在性

Equivariant Allen--Cahn Solutions and the Existence of Cohomogeneity $2$ Minimal Surfaces

研究闭黎曼流形上等变艾伦 - 卡恩解的正则性理论，利用修正方程和相关理论，证明特定条件下解收敛到极小超曲面，还扩展了变分构造，得到等变艾伦 - 卡恩p - 宽度收敛结果，为相关流形存在极小超曲面提供依据。

H-minimal Lagrangian fibrations in Kahler manifolds and minimal Lagrangian vanishing tori in Kahler-Einstein manifolds

H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with negative first Chern class.

On the Lagrangian minimal surface equation and related problems

We give a survey of various existence results for minimal Lagrangian graphs. We also discuss the mean curvature flow for Lagrangian graphs.

Minimal Lagrangian submanifolds in indefinite complex space

Consider the complex linear space C^n endowed with the canonical pseudo-Hermitian form of signature (2p,2(n-p)). This yields both a pseudo-Riemannian and a symplectic structure on C^n. We prove that those submanifolds which are both Lagrangian and minimal with respect to these structures minimize the volume in their Lagrangian homology class. We also describe several families of minimal Lagrangian submanifolds. In particular, we characterize the minimal Lagrangian surfaces in C^2 endowed with its natural neutral metric and the equivariant minimal Lagrangian submanifolds of C^n with arbitrary s…

On the connected components of the moduli space of equivariant minimal surfaces in $\mathbb{CH}^2$

An equivariant minimal surface in $\mathbb{CH}^n$ is a minimal map of the Poincaré disc into $\mathbb{CH}^n$ which intertwines two actions of the fundamental group of a closed surface $Σ$: a Fuchsian representation on the disc and an irreducible action by isometries on $\mathbb{CH}^n$. The moduli space of these can been studied by relating it to the nilpotent cone in each moduli space of $PU(n,1)$-Higgs bundles over the conformal surface corresponding to the map. By providing a necessary condition for points on this nilpotent cone to be smooth this article shows that away from the points corre…

From minimal Lagrangian to J-minimal submanifolds: persistence and uniqueness

Given a minimal Lagrangian submanifold L in a negative Kaehler--Einstein manifold M, we show that any small Kaehler--Einstein perturbation of M induces a deformation of L which is minimal Lagrangian with respect to the new structure. This provides a new source of examples of minimal Lagrangians. More generally, the same is true for the larger class of totally real J-minimal submanifolds in Kaehler manifolds with negative definite Ricci curvature.

Spectral data for Hamiltonian-minimal Lagrangian tori in $CP^2$

In this work, we find spectral data that allow to find Hamiltonian-minimal Lagrangian tori in $CP^2$ in terms of theta functions of spectral curves.

On Hamiltonian-Minimal Lagrangian Tori in CP^2

We obtain some equations for Hamiltonian-minimal Lagrangian surfaces in CP^2 and give their particular solutions in the case of tori.

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

In this paper, we employ the loop group method to study the construction of minimal Lagrangian surfaces in the complex projective plane for which the surface is contractible. We present several new classes of minimal Lagrangian surfaces in $\mathbb{C} P^2$.

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.

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).

Minimal Lagrangian submanifolds in the complex hyperbolic space

In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomegeneity one. We characterize them as the only minimal Lagrangian submanifolds in CH^n foliated by umbilical hypersurfaces of Lagrangian subspaces RH^n of CH^n. Several suitable generalizations of the above construction allow us to get new families of minimal Lagrangian submanifolds in CH^n from curves in CH^1 and (n-1)-dimensional minimal Lagrangian submanifolds of the complex space forms CP^n-1, CH^n-1 and C^n-1.…

A note on the evolution of the Whitney sphere along mean curvature flow

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in $\mathbb{C}^n$ satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.

$f$-minimal Lagrangian Submanifolds in Kähler Manifolds with Real Holomorphy Potentials

The aim of this paper is to study variational properties for $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials. Examples of submanifolds of this kind incuding soliton solutions for Lagrangian mean curvature flow (LMCF). We derive second variation formula for $f$-minimal Lagrangians as a generalization of Chen and Oh's formula for minimal Lagrangians. As a corollary, we obtain stability of expanding and translating solitons for LMCF. We also define calibrated submanifolds with respect to $f$-volume in gradient steady Kähler--Ricci solitons as generalization…

Lagrangian submanifolds in para-complex Euclidean space

We address the study of some curvature equations for distinguished submanifolds in para-Kähler geometry. We first observe that a para-complex submanifold of a para-Kähler manifold is minimal. Next we describe the extrinsic geometry of Lagrangian submanifolds in the para-complex Euclidean space D^n and discuss a number of examples, such as graphs and normal bundles. We also characterize those Lagrangian surfaces of D^2 which are minimal and have indefinite metric. Finally we describe the Lagrangian self-similar solutions of the Mean Curvature Flow which are SO(n)-equivariant.

Hamiltonian-minimal Lagrangian submanifolds in complex space forms

Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one parameter families of embeddings of quotients of certain product manifolds. In addition, new examples of minimal Lagrangian submanifolds in complex projective and hyperbolic spaces also appear. Making use of all of them, we get Hamiltonian-minimal and special Lagrangian cones in complex Euclidean space too.

On a lower bound for the energy functional on a family of Hamiltonian minimal Lagrangian tori in $\mathbb{C}P^2$

We study the energy functional on the set of Lagrangian tori in $\mathbb{C}P^2$ . We prove that the value of the energy functional on a certain family of Hamiltonian minimal Lagrangian tori in $\mathbb{C}P^2$ is strictly larger than energy of the Clifford torus.

A moduli space of minimal affine Lagrangian submanifolds

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Frechet manifold (if it is not the empty set). The moduli space of all connected compact orientable metric Lagrangian embedded surfaces in an almost Kaehler 4-dimensional manifold forms an infinite dimensional Frechet manifold (if it is not the empty set).

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…

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.

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.

$\mathbb{CP}^2$ 中嵌入极小拉格朗日环面的唯一性

Uniqueness of embedded minimal Lagrangian tori in $\mathbb{CP}^2$

本文证明复射影平面中任意嵌入极小拉格朗日环面必全等于克利福德环面，方法为利用五维球面归一化水平提升构造两点函数。

A family of conformally flat Hamiltonian-minimal Lagrangian tori in $\mathbb{CP}^3$

In this paper by reduction we construct a family of conformally flat Hamiltonian-minimal Lagrangian tori in $\mathbb{CP}^3$ as the image of the composition of the Hopf map $\mathcal{H}: \mathbb{S}^7\to \mathbb{CP}^3$ and a map $ψ:\mathbb{R}^3 \to \mathbb{S}^7$ with certain conditions.

On product minimal Lagrangian submanifolds in complex space forms

In this paper we consider minimal Lagrangian submanifolds in $n$-dimensional complex space forms. More precisely, we study such submanifolds which, endowed with the induced metrics, write as a Riemannian product of two Riemannian manifolds, each having constant sectional curvature. As the main result, we give a complete classification of these submanifolds.

Minimal Lagrangian tori in Kahler Einstein manifolds

In this paper we use structure preserving torus actions on Kahler-Einstein manifolds to construct minimal Lagrangian submanifolds. Our main result is: Let N^2n be a Kahler-Einstein manifold with positive scalar curvature with an effective T^n-action. Then precisely one regular orbit L of the T-action is a minimal Lagrangian submanifold of N. Moreover there is an (n-1)-torus T^n-1 in T^n and a sequence of non-flat immersed minimal Lagrangian tori L_k in N, invariant under T^n-1 s.t. L_k locally converge to L (in particular the supremum of the sectional curvatures of L_k and the distance between…

Minimal Lagrangian submanifolds of the complex hyperquadric

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding the geometry of the Lagrangian submanifold at hand. We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface. We also use our techniques to classify all minimal Lagrangian submanifo…

Construction of Hamiltonian-minimal Lagrangian submanifolds in complex Euclidean space

We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of the odd-dimensional unit sphere.

Minimal Lagrangian 2-tori in CP^2 come in real families of every dimension

We show that for every non-negative integer n there is a real n-dimensional family of minimal Lagrangian tori in CP^2, and hence of special Lagrangian cones in C^3 whose link is a torus. The proof utilises the fact that such tori arise from integrable systems, and can be described using algebro-geometric (spectral curve) data.

Reductions of minimal Lagrangian submanifolds with symmetries

Let $M$ be a Fano manifold equipped with a Kähler form $ω\in 2πc_1(M)$ and $K$ a connected compact Lie group acting on $M$ as holomorphic isometries. In this paper, we show the minimality of a $K$-invariant Lagrangian submanifold $L$ in $M$ w.r.t. a globally conformal Kähler metric is equivalent to the minimality of the reduced Lagrangian submanifold $L_0=L/K$ in a Kähler quotient $M_0$ w.r.t. the Hsiang-Lawson metric. Furthermore, we give some examples of Kähler reductions by using a circle action obtained from a cohomogenenity one action on a Kähler-Einstein manifold of positive Ricci curvat…

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…

二重共性问题的正则性及具有大第一贝蒼卫数的球面上的最小超曲面

Regularity of Cohomogeneity two equivariant isotopy minimization problems and minimal hypersurfaces with large first Betti number on spheres

该研究通过证明二重共性等价同调最小化问题的正则性，构造了具有大第一贝蒼衛数的球面上最小超曲面，并展示了其拓扑类型及收敛性质。

Hamiltonian stability of Hamiltonian minimal Lagrangian submanifolds in pseudo- and para-Kähler manifolds

Let L be a Lagrangian submanifold of a pseudo- or para-Kähler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect to Hamiltonian variations. We apply this formula to several cases. In particular we observe that a minimal Lagrangian submanifold L in a Ricci-flat pseudo- or para-Kähler manifold is H-stable, i.e. its second variation is definite and L therefore a local extremizer of the volume with respect to Hamiltonian variations. We also give a stability criterion for sp…

Equivariant minimal surfaces in $\mathbb{CH}^2$ and their Higgs bundles

This paper gives a construction for all minimal immersions $f$ of the Poincaré disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $ρ$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal harmonic map and therefore has a corresponding Higgs bundle. We identify the structure of these Higgs bundles and show how each is determined by properties of the map, including the induced metric and a holomorphic cubic differential on the surface. We show that the moduli space …

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.

On the symmetric equiaffine hyperspheres and the minimal symmetric Lagrangian submanifolds

In this paper, a correspondence via duality is established between the set of locally strongly convex symmetric equiaffine hyperspheres and the set of minimal symmetric Lagrangian submanifolds in a certain complex space form. By using this correspondence theorem, we are able to provide an alternative proof of the classification theorem for the locally strongly convex equiaffine hypersurfaces with parallel Fubini-Pick forms, which has been established recently by Z.J. Hu etc in a totally different way.

On the Lifts of Minimal Lagrangian Submanifolds

We show the total space of the canonical line bundle $\mathbb{L}$ of a Kahler-Einstein manifold $X^n$ supports integrable $SU(n+1)$ structures, or Calabi-Yau structures. The canonical real line bundle $L \subset \mathbb{L}$ over a minimal Lagrangian submanifold $M \subset X$ is calibrated in this setting and hence can be considered as the special Lagrangian lift of $M$. For the integrable $G_2$ and $Spin(7)$ structures on spin bundles and bundles of anti-self-dual 2-forms on self-dual Einstein 4-manifolds constructed by Bryant and Salamon, minimal surfaces with vanishing complex quartic form (…

A family of free boundary minimal surfaces in the unit ball

Using equivariant differential geometry, we provide a family of free boundary minimal surfaces in the unit ball.

Holomorphic vector fields and minimal Lagrangian submanifolds

The purpose of this note is to establish the following theorem: Let N be a Kahler manifold, L be a compact oriented immersed minimal Lagrangian submanifold in N and V be a holomorphic vector field in a neighbourhood of L in N. Let div(V) be the (complex) divergence of V. Then the integral of div(V) over L is 0. Vice versa let N^2n be Kahler-Einstein with non-zero scalar curvature and L^n be a totally real oriented embedded n-dimensional real-analytic submanifold of N s.t. the divergence of any holomorphic vector field defined in a neighbourhood of L in N integrates to 0 on L. Then L is a minim…

Bernstein Type Results for Lagrangian Graphs with Partially Harmonic Gauss Map

We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized.

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…

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。

关于对称空间等变紧致化上的极小有理曲线

Minimal rational curves on equivariant compactifications of symmetric spaces

研究了对称空间等变紧致化上极小有理曲线的轨道闭包性质，证明了高斯非退化性条件下的结论，并推广了完美紧致化结果。

Lagrangian mean curvature flow of Whitney spheres

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.

等变极小最大理论与四维球面的球形伯恩斯坦问题

Equivariant min-max theory and the spherical Bernstein problem in $\mathbb{S}^4$

本文通过等变极小最大理论，在四维球面中构造了非赤道极小超球面，解决了Chern的球形伯恩斯坦问题，并展示了相关问题的解的正则性。

格拉斯曼流形与旗流形的最小等变嵌入

Minimal equivariant embeddings of the Grassmannian and flag manifold

本文证明旗流形（含格拉斯曼流形）存在维数为$(n-1)(n+2)/2$的SO_n等变嵌入，该维数最小且所有最小维等变嵌入均等价。

Flows of constant mean curvature tori in the 3-sphere: The equivariant case

We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov embedded tori therein. We conclude with an instability result.

Lagrangian submanifolds attaining equality in the improved Chen's inequality

Recently Oprea gave an improved version of Chen's inequality for Lagrangian submanifolds of $\mathbb CP^n(4)$. For minimal submanifolds this inequality coincides with the original previously proved version. We consider here those non minimal 3-dimensional Lagrangian submanifolds in $\mathbb CP^3 (4)$ attaining at all points equality in the improved Chen inequality. We show how all such submanifolds may be obtained starting from a minimal Lagrangian surface in $\mathbb CP^2(4)$ and a suitable horizontal curve in $S^3(1)$.