Evolution Equations for Special Lagrangian 3-Folds in C3

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C ^m . The previous paper (Math. Ann. 320 (2001), 757–797), defined the idea of evolution data, which includes an (m – 1)-submanifold P in R ⁿ , and constructed a family of special Lagrangian m-folds N in C ^m , which are swept out by the image of P under a 1-parameter family of affine maps _t : R ⁿ C ^m , satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C³. We find a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C³.Our main results are a number of new families of special Lagrangian 3-foldsin C³, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R³ or ¹× R². Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind.We hope these 3-folds will be helpful in understanding singularities ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.     

Special Lagrangian Submanifolds with Isolated Conical Singularities. I. Regularity

This is the first in a series of five papers studying special Lagrangian submanifolds(SLV m-folds) X in almost Calabi–Yau m-folds M with singularitiesx ₁, ..., x _n locally modelled on special Lagrangiancones C ₁, ..., C _n in ^m with isolated singularities at 0. Readers are advised to begin with Paper V.This paper lays the foundations for the series, giving definitions and provingauxiliary results in symplectic geometry and asymptotic analysis that will be needed later.It also proves results on the regularity of X near its singular points.We show that X converges to the cone C _i near x _i with all its derivatives,at rates determined by the eigenvalues of the Laplacian on C _i .We show that if X is a special Lagrangian integral current with a tangent cone C at x satisfying some conditions, then X has an isolated conical singularity at x in our sense. We also prove analogues of many of our results for Asymptotically Conical SL m-folds in ^m .     

Special Lagrangian Submanifolds with Isolated Conical Singularities. II. Moduli spaces

This is the second in a series of five papers studying special Lagrangiansubmanifolds (SLV m-folds) X in (almost) Calabi–Yau m-folds M with singularities x ₁ , ..., x _n locally modelled on specialLagrangian cones C ₁, ..., C _n in ^m with isolated singularities at 0.Readers are advised to begin with Paper V.This paper studies the deformation theory of compact SL m-folds X in Mwith conical singularities. We define the moduli space _X of deformations of X in M, and construct a natural topology on it. Then we show that _X is locally homeomorphic to the zeroes of a smooth map : _{X X} between finite-dimensional vector spaces.Here the infinitesimal deformation space _X depends only on the topology of X, and the obstruction space _X only on the cones C ₁, ..., C _n at x ₁, ..., x _n . If the cones C _i are stable then _X is zero, and _X is a smooth manifold. We also extend our results to families of almost Calabi–Yau structures on M.     

On Hamiltonian-Minimal Lagrangian Tori in $$\mathbb{C}P^2 $$

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

On second order minimal Lagrangian sub manifolds in complex space forms

In this paper, we determine all second order minimal Lagrangian submanifolds in complex space forms whose cubic forms have the largest non-trivial continuous symmetries. We describe these minimal Lagrangian submanifolds from the viewpoint of Bryant and study their geometric properties.     

Special Lagrangian Submanifolds with Isolated Conical Singularities. IV. Desingularization,Obstructions and Families

This is the fourth in a series of five papers studying special Lagrangian submanifolds(SLV m-folds) X in (almost) Calabi–Yau m-folds M with singularities x ₁,..., x _n locally modelled on special Lagrangian cones C ₁,..., C _n in ^m with isolated singularities at 0. Readers are advised to begin with Paper V.Paper III and this one construct desingularizations of X, realizing X as a limitof a family of compact, nonsingular SL m-folds C ^t in M for small t > 0. Suppose L ₁,..., L _n are Asymptotically Conical SL m-folds in ^m, withL _i asymptotic to the cone C _i at infinity. We shrink L _i by a small t > 0, and gluetL _i into X at x _i for i= 1,..., n to get a 1-parameter family of compact, nonsingularLagrangian m-folds N ^t for small t> 0.Then we show using analysis that when t is sufficiently small we can deform N ^t toa compact, nonsingular special Lagrangian m-fold C ^t, via a small Hamiltonian deformation. This C ^t depends smoothly on t, and as t 0 it converges to the singular SL m-fold X, in the sense of currents.Paper III studied simpler cases, where by topological conditions on X and L _i we avoid obstructions to the existence of C ^t. This paper considers more complex cases when theseobstructions are nontrivial, and also desingularization in families of almost Calabi–Yaum-folds M ^s for sF, rather than in a single almost Calabi–Yau m-fold M.     

Special Lagrangian Submanifolds with Isolated Conical Singularities. III. Desingularization,The Unobstructed Case

This is the third in a series of five papers studying special Lagrangian submanifolds(SLV m-folds) X in (almost) Calabi–Yau m-folds M with singularities x ₁, ..., x _n locally modelled on special Lagrangian cones C ₁, ..., C _n in ^m with isolated singularities at 0. Readers are advised to begin with Paper V.This paper and Paper IV construct desingularizations of X, realizing X as a limitof a family of compact, nonsingular SL m-folds ^t in M for small t > 0.Suppose L ₁, ..., L _n are Asymptotically Conical SL m-folds in ^m , with L _i asymptotic to the cone C _i at infinity. We shrink L _i by a small t > 0, and glue tL _i into X at x _i for i= 1, ..., nto get a 1-parameter family of compact, nonsingular Lagrangian m-folds N ^t for small t > 0.Then we show using analysis that when t is sufficiently small we can deform N ^t to a compact,nonsingular special Lagrangian m-fold ^t , via a small Hamiltonian deformation.This ^t depends smoothly on t, and as t 0 it converges to the singular SL m-fold X, in the sense of currents.This paper studies the simpler cases, where by topological conditions on X and L _i we avoid various obstructions to existence of ^t . Paper IV will consider more complex cases when these obstructions are nontrivial, and also desingularization in families of almost Calabi–Yau m-folds.     

New construction of special Lagrangian submanifolds in T^＊R^n equipped with the standard metric

A class of twisted special Lagrangian submanifolds in T~*R~n and a kind of austere submanifold from conormal bundle of minimal surface of R~3 are constructed.     

Pseudo-parallel Lagrangian submanifolds are semi-parallel

We prove a conjecture formulated by Pablo M. Chacon and Guillermo A. Lobos in [P.M. Chacon, G.A. Lobos, Pseudo-parallel Lagrangian submanifolds in complex space forms, Differential Geom. Appl. 27 (1) (2009) 137–145, doi:10.1016/j.difgeo.2008.06.014] stating that every Lagrangian pseudo-parallel submanifold of a complex space form of dimension at least 3 is semi-parallel. We also propose to study another notion of pseudo-parallelity which is more adapted to the Kaehlerian setting.     

Equivariant Lagrangian Minimal S^3 in CP^3

The equivariant Lagrangian minimal immersion of Sa into C.Pa is studied. The complete classification and analytic expression for such kinds of immersions are given.     

New construction of special Lagrangian submanifolds in <Emphasis Type="Italic">T</Emphasis>*R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> equipped with the standard metric

A class of twisted special Lagrangian submanifolds in T＊R^n and a kind of austere submanifold from conormal bundle of minimal surface of R^3 are constructed.     

Construction of Lagrangian Submanifolds in Complex Hyperquadric

In this paper, the authors present a method to construct the minimal and H-minimal Lagrangian submanifolds in complex hyperquadric Q_n from submanifolds with special properties in odd-dimensional spheres. The authors also provide some detailed examples.     

Isometric Embeddings of Families of Special Lagrangian Submanifolds

We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi–Yau manifolds. For example, we prove that given any real-analytic one parameter family of Riemannian metrics g _t on a three-dimensional manifold Y with volume form independent of t and with a real-analytic family of nowhere vanishing harmonic one forms θ _t , then (Y,g _t ) can be realized as a family of special Lagrangian submanifolds of a Calabi–Yau manifold X. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus action. We use this to construct examples of n-parameter families of special Lagrangian tori inside n + k-dimensional Calabi–Yau manifolds with torus symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian fibrations with T ²-symmetry. Mathematics Subject Classification (2000): 53-XX, 53C38.Communicated by N. Hitchin (Oxford)     

Deformation of special Lagrangian suborbifolds

The purpose of this paper is to generalize the deformation theory of special Lagrangian submanifolds developed by Mclean and Hitchin to special Lagrangian suborbifolds.     

Locally symmetric minimal affine Lagrangian surfaces in C<Superscript>2</Superscript>

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry. The research supported by the KBN grant 1 PO3A 034 26.     

Calibrated Embeddings in the Special Lagrangian and Coassociative Cases

Every closed, oriented, real analytic Riemannian3–manifold can be isometrically embedded as a specialLagrangian submanifold of a Calabi–Yau 3–fold, even as thereal locus of an antiholomorphic, isometric involution. Every closed,oriented, real analytic Riemannian 4–manifold whose bundle of self-dual2–forms is trivial can be isometrically embedded as a coassociativesubmanifold in a G₂-manifold, even as the fixed locus of ananti-G₂ involution.These results, when coupledwith McLean's analysis of the moduli spaces of such calibratedsubmanifolds, yield a plentiful supply of examples of compact calibratedsubmanifolds with nontrivial deformation spaces.     

Classification of Casorati ideal Lagrangian submanifolds in complex space forms

In this paper, using optimization methods on Riemannian submanifolds, we establish two improved inequalities for generalized normalized δ-Casorati curvatures of Lagrangian submanifolds in complex space forms. We provide examples showing that these inequalities are the best possible and classify all Casorati ideal Lagrangian submanifolds (in the sense of B.-Y. Chen) in a complex space form. In particular, we generalize the recent results obtained in G.E. Vîlcu (2018) [34].