Lagrangian Surfaces Research Articles

Basic Remarks on Lagrangian Submanifolds of Hyperkähler Manifolds

Abstract This note presents basic restrictions on the topology of Lagrangian surfaces of hyper-Kähler $4$-folds and a remark on the interaction of a Lagrangian subvariety of a hyper-Kähler variety with a Lagrangian fibration of the latter.

Light curves from symmetric, polytropic contact binaries

ABSTRACT We study the structure of contact binaries assuming a polytropic relation between pressure and density, restricting ourselves to the case of equal-mass components, i.e. symmetric contact binaries. In this case, matter is at rest in the corotating reference frame making this problem far simpler than the general case of non-symmetric contact binaries. We compute these structures assuming values of the polytropic index of n = 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 employing a self-consistent technique due to Hachisu. As a part of the results, we find the shape of their surfaces. While for the case of n = 3.5, such surfaces are very close to those corresponding to equipotentials of the restricted Lagrangian three-body problem, for lower n values the departure is remarkable. We propose a generalized function to fit these surfaces, which allows us to perform an accurate integration of the light curve due to the object. Then, for values of n > 0.0 we computed a family of light curves considering different inclinations and values for the width of the neck connecting the components (or equivalently, the fillout factor) of the pair. We compare our calculations with the solution found for the symmetric contact binary V803 Aquilae by employing the popular phoebe code, that assumes the above-mentioned Lagrangian equipotential surfaces. We conclude that considering polytropic structures leads to parameters that may be appreciably different from those deduced by assuming that their surfaces correspond to equipotentials of the restricted three-body problem.

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

We construct an explicit map from a generic minimal δ(2)-ideal Lagrangian submanifold of Cn to the quaternionic projective space HPn−1, whose image is either a point or a minimal totally complex surface. A stronger result is obtained for n=3, since the above mentioned map then provides a one-to-one correspondence between minimal δ(2)-ideal Lagrangian submanifolds of C3 and minimal totally complex surfaces in HP2 which are moreover anti-symmetric. Finally, we also show that there is a one-to-one correspondence between such surfaces in HP2 and minimal Lagrangian surfaces in CP2.

Braid loops with infinite monodromy on the Legendrian contact DGA

We present the first examples of elements in the fundamental group of the space of Legendrian links in ( S 3 , ξ st ) $(\mathbb {S}^3,\xi _{\text{st}})$ whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These new families include the first-known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our examples to construct other links with infinitely many fillings, and in particular give the first Floer-theoretic proof that Legendrian ( n , m ) $(n,m)$ torus links have infinitely many Lagrangian fillings if n ⩾ 3 , m ⩾ 6 $n\geqslant 3,m\geqslant 6$ or ( n , m ) = ( 4 , 4 ) , ( 4 , 5 ) $(n,m)=(4,4),(4,5)$ . In addition, for any given higher genus, we construct a Weinstein 4-manifold homotopic to the 2-sphere whose wrapped Fukaya category can distinguish infinitely many exact closed Lagrangian surfaces of that genus in the same smooth isotopy class, but distinct Hamiltonian isotopy classes. A key technical ingredient behind our results is a new combinatorial formula for decomposable cobordism maps between Legendrian contact DGAs with integer (group ring) coefficients.

Lagrangian surfaces of constant angle in the complex Euclidean plane

Lagrangian surfaces of constant angle in the complex Euclidean plane

Refined disk potentials for immersed Lagrangian surfaces

We define a refined Gromov–Witten disk potential of monotone immersed Lagrangian surfaces in a symplectic 4‑manifold that are self-transverse as an element in a capped version of the Chekanov–Eliashberg dg‑algebra of the singularity links of the double points (a collection of Legendrian Hopf links). We give a surgery formula that expresses the potential after smoothing a double point. We study refined potentials of monotone immersed Lagrangian spheres in the complex projective plane and find monotone spheres that cannot be displaced from complex lines and conics by symplectomorphisms. We also derive general restrictions on sphere potentials using Legendrian lifts to the contact 5‑sphere.

Infinitely many Lagrangian fillings

We prove that all maximal-tb positive Legendrian torus links $(n,m)$ in the standard contact $3$-sphere, except for $(2,m)$, $(3,3)$, $(3,4)$ and $(3,5)$, admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances that induce faithful actions of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ and the mapping class group $M_{0,4}$ into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a $2$-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.

Rigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary

In this note, we study minimal Lagrangian surfaces in B4 with Legendrian capillary boundary on S3. On the one hand, we prove that any minimal Lagrangian surface in B4 with Legendrian free boundary on S3 must be an equatorial plane disk. On the other hand, we show that any annulus type minimal Lagrangian surface in B4 with Legendrian capillary boundary on S3 must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng [12].

Biharmonic submanifolds in metallic Riemannian manifolds

In this paper, we study the biharmonic submanifolds of Riemannian manifolds endowed with metallic and complex metallic structures. In case of both the structures, we obtain the necessary and sufficient conditions for a submanifold to be biharmonic. Particularly, we find the estimates for mean curvature of Lagrangian and complex surfaces.

On the compactness of Hamiltonian stationary Lagrangian surfaces in Kähler surfaces

We prove a bubble tree convergence theorem for a sequence of closed Hamiltonian stationary Lagrangian surfaces with bounded areas and Willmore energies in a complete Kahler surface. We also prove two strong compactness theorems on the space of Hamiltonian stationary Lagrangian tori in $${\mathbb {C}}^2$$ and $${{\mathbb {C}}}{{\mathbb {P}}}^2$$ respectively.

Minimal Lagrangian surfaces in [formula omitted] 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 domain of the surface is contractible. We present several new classes of minimal Lagrangian surfaces in ℂP2.

An energy gap phenomenon for the Whitney sphere

In this paper, we study Lagrangian surfaces satisfying nabla ^*T=0 , where T=-2nabla ^*(check{A}lrcorner omega ) and check{A} is the Lagrangian trace-free second fundamental form. We obtain a gap lemma for such a Lagrangian surface.

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 form. Moreover, for each timelike Lagrangian surface in \(\mathbb {CH}^{2}_1\) we define natural Gauss maps into certain homogeneous spaces and prove a Ruh–Vilms-type theorem, characterizing timelike minimal Lagrangian surfaces among all timelike Lagrangian surfaces in terms of the harmonicity of these Gauss maps.

Lagrangian surfaces with Legendrian boundary

In this paper, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces. Then we present several interesting examples of Lagrangian submanifolds satisfying this boundary condition and we prove a Lagrangian version of the Nitsche (or Hopf) type theorem. Some problems are proposed at the end of this paper.

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.

Nonorientable Lagrangian surfaces in rational 4–manifolds

We show that for any nonzero class $A$ in $H_2(X; \mathbb{Z}_2)$ in a rational 4-manifold $X$, $A$ is represented by a nonorientable embedded Lagrangian surface L (for some symplectic structure) if and only if $P(A)\equiv (L) (mod\ 4)$; where $P(A)$ denotes the mod 4 valued Pontrjagin square of $A$.

On the Lagrangian Angle and the Kähler Angle of Immersed Surfaces in the Complex Plane ℂ2

In this paper, we discuss the Lagrangian angle and the Kahler angle of immersed surfaces in ℂ2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in ℂ2 than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant Kahler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant Kahler angle is generally non-vanishing. Secondly, we obtain two pinching results for the Kahler angle which imply rigidity theorems of self-shrinkers with Kahler angle under the condition that $${\smallint _M}{\left| h \right|^2}{{\rm{e}}^{ - {{{{\left| x \right|}^2}} \over 2}}}{\rm{d}}{V_M}\; < \;\infty $$ , where h and x denote, respectively, the second fundamental form and the position vector of the surface.

Lagrangian Bonnet Problems in Complex Space Forms

In this note we consider Lagrangian Bonnet problem for Lagrangian surfaces in complex space forms. We first give a Bonnet type 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 Lagrangian Bonnet surfaces in complex space forms, and obtain some characterizations of such surfaces.

Survey on real forms of the complex A2(2)-Toda equation and surface theory

AbstractThe classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k-symmetric spaces over reductive Lie groups, [8].In this survey we will show that to each of the five different types of real forms for a loop group of A2(2) there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in ℂℙ2, minimal Lagrangian surfaces in ℂℍ2, timelike minimal Lagrangian surfaces in ℂℍ12, proper definite affine spheres in ℝ3 and proper indefinite affine spheres in ℝ3, respectively.

A Lagrangian vertical coordinate version of the ENDGame dynamical core. Part I: Formulation, remapping strategies, and robustness

Previous work provides evidence that Lagrangian conservation and related properties of a numerical model dynamical core can be improved by the use of a Lagrangian or quasi‐Lagrangian vertical coordinate (LVC). Most previous model developments based on this idea have made the hydrostatic approximation. Here the LVC is implemented in a non‐hydrostatic compressible Euler equation dynamical core using almost identical numerical methods to ENDGame, the operational dynamical core of the Met Office atmospheric Unified Model. This enables a clean comparison of LVC and height‐coordinate versions of the dynamical core using numerical methods that are as similar as possible. Since Lagrangian surfaces distort over time, model level heights are continually reset to certain “target levels” and the values of model fields are remapped onto their new locations. Different choices for these target levels are discussed, along with remapping strategies that focus on different conservation or balance properties. Sample results from a baroclinic instability test case are presented. The LVC formulation is found to be rather less robust than the height‐coordinate version; some reasons for this are discussed.