Ildefonso Castro
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.
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A curvature-ellipse-based Monge normal form of a regular surface in $\mathbb{R}^4$ and its applications
本文提出基于曲率椭圆主轴的 $\mathbb{R}^4$ 中曲面 Monge 标准形，仅用欧氏运动和局部重参数化保持曲率椭圆不变量，并应用于 $2$-jet 几何与正交投影几何。
At each point in an immersed surface in $\mathbb R^4$ there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. More recently, at the singular point of a corank 1 singular surface in $\mathbb R^3$, a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in $\mathbb R^4$ to $\mathbb R^3$ in a tangent direction corank 1 singularities appear generically. The projection has a cross-cap singularity unless the direction of projection is asymptotic, where m…
At any point of a surface in the four-dimensional Euclidean space we consider the geometric configuration consisting of two figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of the two geometric figures. We give some non-trivial examples of surfaces from the classes in consideration.
We consider smooth plane curves which are convex with respect to the origin. We describe centro-affine invariants (that is, GL_+(2,R)-invariants), such as centro-affine curvature and arc length, in terms of the canonical Lorentz structure on the three dimensional space of all the ellipses centered at zero, by means of null curves of osculating ellipses. This is the centro-affine analogue of the approach to conformal invariants of curves in the sphere introduced by Langevin and O'Hara, using the canonical pseudo Riemannian metric on the space of circles.
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.
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.
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。
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.
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.
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.
We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in $\mathbb C^{n}$.
We provide several rigidity results for the Clifford torus in the class of compact self-shrinkers for Lagrangian mean curvature flow.
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。
We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the 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
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。
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.
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.
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.
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。
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.
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.
We introduce Lagrangian mean curvature flow with boundary in Calabi--Yau manifolds by defining a natural mixed Dirichlet-Neumann boundary condition, and prove that under this flow, the Lagrangian condition is preserved. We also study in detail the flow of equivariant Lagrangian discs with boundary on the Lawlor neck and the self-shrinking Clifford torus, and demonstrate long-time existence and convergence of the flow in the first instance and of the rescaled flow in the second.
In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold is Lagrangian and almost calibrated by ReΩin a Calabi-Yau n-fold (M,Ω), and T>0 is the first blow-up time of the mean curvature flow, then the tangent cone of the mean curvature flow at a singular point (X,T) is a stationary Lagrangian integer multiplicity current in R\sup 2n with volume density greater than one at X. When n=2, the tangent cone consists of a finite union of more than one 2-planes in R\sup 4 which are complex in a complex structure on R\su…
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。
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。
We study geometric properties of the Lagrangian self-shrinking tori in $\mathbb R^4$. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori and then combining with the compactness theorem in \cite{CMa}. When the area bound is small, we show that any Lagrangian self-shrinking torus in $\mathbb R^4$ with small area is embedded with uniform curvature estimates, and the space of such tori is com…
Mean Curvature Flows of Two-Convex Lagrangians
本文证明二凸拉格朗日平均曲率流的正则性、全局存在性与收敛性，利用新发现的单调量控制二凸性，并通过酉变换推广至面积递减情形，显著改进了凸情形的已知结果。
We consider the evolution by mean curvature flow of Lagrangian submanifolds of the complex projective space CP^n. We prove that, if the initial value satisfies a suitable pinching condition, then the flow exists for all times and the manifold converges to a totally geodesic submanifold. As a corollary, we obtain that a Lagrangian submanifold satisfying our pinching condition is diffeomorphic to a real projective space.
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.
The self-similar solutions to the mean curvature flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asymptotic behavior for the singularities of the mean curvature flows. We also extend the results on special Lagrangian submanifolds on $\mathbb C^n$ to the toric Calabi-Yau cones over Sasaki-Einstein manifolds.
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。
In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifol…
In this paper, we construct finite blow-up examples for symplectic mean curvature flows and we study properties of symplectic translating solitons. We prove that, the Kähler angle $α$ of a symplectic translating soliton with $\max |A|=1$ satisfies that $\sup |α|>\fracπ{4}\frac{|T|}{|T|+1}$ where $T$ is the direction in which the surface transltes.
Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class
本文针对零Maslov类拉格朗日平均曲率流的II型奇点，给出其爆破极限的必要条件，并尝试解答Joyce-Lee-Tsui构造的平移孤立子能否作为该流爆破极限的开放问题。
Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature flow in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones \cite{JLT} in dimension two. The simplest (non trivial) example in our family is characterized as the only (non totally geodesic) Hamiltonian stationary Lagrangian translating soliton for mean curvature flow in complex Euclidean plane.
Ancient solutions and translators of Lagrangian mean curvature flow
本文研究复空间中Lagrange平均曲率流的近校准精确古老解，通过吹缩为两个相交Lagrange平面的静态并集这一条件证明其必为平移子，并在$\mathbb{C}^2$中对熵小于3的情形给出分类。
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。
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.
We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of special Lagrangians or Lagrangian minimal surfaces in the variational approach. It is known that these cone singularities cannot be resolved by any smooth oriented Lagrangian submanifolds. The shrinkers and expanders that we found can be glued together to yield solutions of the Brakke motion-a wea…
From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the $18$seventies), and what are the principal first normal directions, (first studied by Kostadin Trencevski in the $19$nineties), and what are their corresponding Casorati curvatures. For reasons of simplicity of exposition only, hereafter this will merely be done explicitly in the case of arbitrary submanifolds in Euclidean spaces. Then, for the special case of Lagrangian submanifolds in complex Euclidean spaces, the natural relationships betwe…
We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time $t=0$. In particular, under the mean curvature flow the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as $t\t…
We consider self-similar solutions to mean curvature evolution of entire Lagrangian graphs. When the Hessian of the potential function $u$ has eigenvalues strictly uniformly between -1 and 1, we show that on the potential level all the shrinking solitons are quadratic polynomials while the expanding solitons are in one-to-one correspondence to functions of homogenous of degree 2 with the Hessian bound. We also show that if the initial potential function is cone-like at infinity then the scaled flow converges to an expanding soliton as time goes to infinity.
Singularity formations in Lagrangian mean curvature flow
该文研究拉格朗日平均曲率流中由重数1的光滑特殊拉格朗日锥给出切流的奇点，得到多类唯一性与爆破极限存在性结果，还推广了Lawlor neck的唯一性定理，并应用于横截平面切流奇点的建模。
Generic neck pinch singularities along 2D Lagrangian mean curvature flow
本文研究了二维拉格朗日均曲率流中的非退化颈部 pinch 神经奇点，证明其稳定性并回答了相关问题，同时引入了无法发生的非退化泪滴状奇点。
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.
In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a Kähler-Einstein manifold to more general Kähler manifolds including a Fano manifold equipped with a Kähler form $ω\in 2πc_1(M)$ by using the methodology proposed by T. Behrndt. Namely, we first consider a weighted measure on a Lagrangian submanifold $L$ in a Kähler manifold $M$ and investigate the variational problem of $L$ for the weighted volume functional. We call a stationary point of the weighted volume functional $f$-minimal, and define the notion of Hami…
Let $M$ be an $n$-dimensional Lagrangian submanifold of a complex space form. We prove a pointwise inequality $$δ(n_1,\ldots,n_k) \leq a(n,k,n_1,\ldots,n_k) \|H\|^2 + b(n,k,n_1,\ldots,n_k)c,$$ with on the left hand side any delta-invariant of the Riemannian manifold $M$ and on the right hand side a linear combination of the squared mean curvature of the immersion and the constant holomorphic sectional curvature of the ambient space. The coefficients on the right hand side are optimal in the sence that there exist non-minimal examples satisfying equality at at least one point. We also character…
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 the "twistor energy" by means of the associated twistor family (i.e. 2-sphere of complex structures). We will show that the mean curvature flow starting at any hyper-Lagrangian submanifold with sufficiently small twistor energy will exist for all time and converge to a complex La…
Curvature and Lagrangian submanifolds of the homogeneous nearly Kähler $\mathbb{C}P^3$
本文通过 Hopf 纤维定义 homogeneous nearly Kähler 结构，并研究了 $\mathbb{C}P^3$ 中 Lagrangian 子流形的曲率性质与等距群。
Let $F_n :(Σ, h_n) \to \mathbb C^2$ be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics $\{h_n\}$ converges smoothly to a Riemannian metric $h$. We show that a subsequence of $\{F_n\}$ converges smoothly to a branched conformally immersed Lagrangian self-shrinker $F_\infty : (Σ, h)\to \mathbb C^2$. When the area bound is less than $16π$, the limit $F_\infty$ is an embedded torus. When the genus of $Σ$ is one, we can drop the assumption on convergence $h_n\to h$. When the genus of $Σ…
We consider the complex hyperbolic quadric ${Q^*}^n$ as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on ${Q^*}^n$, which are then used to define local angle functions on any Lagrangian submanifold of ${Q^*}^n$. We prove that a Lagrangian immersion into ${Q^*}^n$ can be seen as the Gauss map of a spacelike hypersurface of (real) anti-de Sitter space and relate the angle functions to the principal curvatures of this hypersurface. We also give a formula relating the mean curvature of the Lagran…
The indicatrix or curvature ellipse and the characteristic curve of a surface in $\mathbf R^4$ are presented, as well as the projective duality connecting them. The characterisation of points in the surfaces as elliptic, parabolic and hyperbolic points, and the inflection points, are also discussed.
Entire solutions of two-convex Lagrangian mean curvature flows
本文研究由整体函数梯度图构成的拉格朗日子流形在平均曲率流下的演化，在较弱的二凸性条件下证明了特殊拉格朗日演化方程的长时间存在性与收敛性，推广了以往仅适用于强凸情形的结果。
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…
It is classically known that closed geodesics on a compact Riemann surface with a metric of negative curvature strictly minimize length in their free homotopy class. We'd like to generalize this to Lagrangian submanifolds in Kähler manifolds of negative Ricci curvature. The only known result in this direction is a theorem on Y.I. Lee for certain Lagrangian submanifolds in a product of two Riemann surfaces of constant negative curvature. We develop an approach to study this problem in higher dimensions. Along the way we prove some weak results (volume-minimization outside of a divisor) and give…
We construct new examples of self-similar solutions and translating solitons for Lagrangian mean curvature flow by extending the method of Joyce, Lee and Tsui. Those examples include examples in which the Lagrangian angle is arbitrarily small as the examples of Joyce, Lee and Tsui.
In this paper, we derive a priori interior Hessian estimates for Lagrangian mean curvature equation if the Lagrangian phase is supercritical and has bounded second derivatives.
Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II
本文研究伪欧氏空间中整体拉格朗日图的平均曲率流，证明介于 Monge-Ampère 型方程与特殊拉格朗日抛物方程之间的非线性抛物方程存在光滑解，并给出高阶导数界、衰减估计及收敛到光滑自膨胀解的结果。
Regularity for convex viscosity solutions of Lagrangian mean curvature equation
本文研究Lagrangian平均曲率方程凸粘性解的正则性，在相位二阶导数Hölder连续的条件下证明其解为正则解。
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