arXiv research

Construct minimal Lagrangian surfaces in complex projective plane via loop group method.

Study minimal Lagrangian surfaces in complex projective plane, focusing on contractible cases.

Study confirms conjecture: minimal Lagrangian surfaces with Legendrian boundary are rigid.

Derives Lagrangian for minimal surfaces, proving tangential variations vanish.

Maps Lagrangian submanifolds to quaternionic projective space, proving one-to-one correspondences.

Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.

Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid.

Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.

The paper studies Lagrangian surfaces in a specific Riemannian product space.

The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagran…

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 $\mathbb {CH}^2$…

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.

We introduce the notion of a minimal Lagrangian connection on the tangent bundle of a manifold and classify all such connections in the case where the manifold is a compact oriented surface of non-vanishing Euler characteristic. Combining our classification with results of Labourie and Loftin, we conclude that every pr…

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.

Develops a new method for minimal Lagrangian surfaces in complex quadrics.

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…

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 m…

Following earlier work of Loftin-McIntosh, we study minimal Lagrangian immersions of the universal cover of a closed surface (of genus at least 2) into CH2, with prescribed data of a conformal structure plus a holomorphic cubic differential. We show existence and non-uniqueness of such minimal Lagrangian immersions. We…

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

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 t…

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

We use an elliptic differential equation of Tzitzeica type to construct a minimal Lagrangian surface in CH2 from the data of a compact hyperbolic Riemann surface and a small holomorphic cubic differential. The minimal Lagrangian surface is invariant under an SU(2,1) action of the fundamental group. We further parameter…

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 th…

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.

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 closel…

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.

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 exa…

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…

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 La…

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

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 Lagr…

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 theo…

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 derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.

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-isot…

Explains Bernstein theorems for various geometric PDEs.

In this paper we suggest a method for constructing minimal Lagrangian immersions of $R^2$ in $CP^2$ with induced diagonal metric in terms of Baker-Akhiezer functions of algebraic 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, Hami…

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…

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 …

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 …

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 p…

A surface $Σ\subset S^5 \subset \mathbb{C}^3$ is called \emph{special Legendrian} if the cone $0 \times Σ\subset \mathbb{C}^3$ is special Lagrangian. The purpose of this paper is to propose a general method toward constructing compact special Legendrian surfaces of high genus. It is proved \emph{there exists a compact,…

The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups 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 spa…

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

Paper studies complex Lagrangian surfaces and their relation to $\mathrm{SL}(3,\mathbb{C})$-representations.

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 exampl…

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…