Study gluing and deformations of special Lagrangian submanifolds.
problem Deforming special Lagrangian submanifolds in Calabi-Yau manifolds.
method Proves well-defined gluing map and local diffeomorphism for deformations.
result Gluing map defines local diffeomorphism between deformations.
We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth sp…
Study on deformations of special Lagrangians with boundary in Calabi-Yau manifolds.
problem Deformation problem for special Lagrangians with boundary constraints.
method Identifying tangent vectors with harmonic 1-forms vanishing on the boundary, proving unobstructed deformations.
result Moduli space of special Lagrangians with boundary is a smooth manifold.
The study introduces flows for lagrangian varifolds to converge to special lagrangian cycles.
problem Understanding the behavior of lagrangian cycles in Calabi-Yau manifolds.
method Constructing mass-decreasing flows of lagrangian varifolds and cycles with vanishing Maslov index.
result Special lagrangian cycles are generated by the flow of cycles, and they represent a part of the lagrangian homology.
New special Lagrangian submanifolds with multiple conical singularities found.
problem Existence of special Lagrangian submanifolds with multiple conical singularities.
method Extending Caffarelli-Hardt-Simon perturbation argument to special Lagrangian setting and proving a bridge principle.
result Existence of conically singular special Lagrangian submanifolds with prescribed regular tangent cones.
We introduce special Lagrangian submanifolds in C^m and in (almost) Calabi-Yau manifolds, and survey recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. The paper is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the sub…
We study special Lagrangian cones in $\C^n$ with isolated singularities. Our main result constructs an infinite family of special Lagrangian cones in $\C^3$ each of which has a toroidal link. We obtain a detailed geometric description of these tori. We prove a regularity result for special Lagrangian cones in $\C^3$ wi…
Quantifies closeness of special Lagrangians under Floer conditions.
problem Estimating closeness of special Lagrangians.
method Floer theoretic conditions leading to quantitative estimates.
result Strong-weak uniqueness theorem for special Lagrangians.
Construct special Lagrangian pair of pants in n dimensions.
problem Constructing special Lagrangian pair of pants in general dimensions.
method Inside the cotangent bundle of Tn with the Euclidean structure. result Special Lagrangian pair of pants constructed in general dimensions.
In this paper, which is a natural continuation of our previous paper math.DG/0504557, we describe some special Lagrangians of cohomogeneity one in the resolved conifold. Our main result gives a foliation of the resolved conifold by T^2-invariant special Lagrangians, where the generic leaf is topologically T^2 X R. We a…
Study special Lagrangian moduli spaces with boundary.
problem Understanding geometric structures on moduli spaces of special Lagrangians.
method Investigates geometric structures and constructs special affine structures and a Hessian metric.
result Constructs a pair of special affine structures and a Hessian metric on the moduli space.
Study characterizes SU(3)-structures with special Lagrangian 3-folds as minimal.
problem Characterizing SU(3)-structures with special Lagrangian 3-folds.
method Derivation of mean curvature formulas and analysis of SU(3)-structures.
result Obtained formulas for mean curvature and characterized SU(3)-structures.
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
Constructs special Lagrangian 3-spheres in non-Kähler compact threefolds.
problem Understanding transitions between Kähler and non-Kähler geometries.
method Analyzes topological transitions of Calabi-Yau threefolds to construct special Lagrangian cycles.
result Special Lagrangian 3-spheres emerge from non-Kähler geometries, exchanging holomorphic 2-cycles for 3-cycles.
Tropical curves match to special Lagrangian shapes.
problem Connecting tropical geometry to special Lagrangian shapes.
method Gluing construction that matches tropical local models to Lagrangian shapes.
result Locally planar tropical curves can be realized as special Lagrangian limits.
The purpose of this paper is to give an application of the gluing theorem for special Lagrangian submanifolds of a Calabi-Yau 3-fold. We proved a gluing theorem before to smooth a codimension-two singularity of a particular special Lagrangian submanifold. In this paper we will show that this theorem can be applied to m…
This is the first in a series of papers on special Lagrangian submanifolds in C^m. We study special Lagrangian submanifolds in C^m with large symmetry groups, and give a number of explicit constructions. Our main results concern special Lagrangian cones in C^m invariant under a subgroup G in SU(m) isomorphic to U(1)^{m…
Real analytic solutions found for special Lagrangian equation.
problem Analyzing convex solutions of the special Lagrangian equation.
method Interior regularity established for convex viscosity solutions.
result All convex solutions are real analytic in the interior.
New Lagrangian and special Lagrangian examples found in complex space.
problem Constructing exact Lagrangian and special Lagrangian submanifolds with symmetries.
method Using an Ansatz generalizing Castro-Lerma's construction, with admissible compact and non-compact subgroups.
result Explicit examples of Lagrangian translators and special Lagrangians with various symmetries.
The paper proves transversality for special Lagrangian submanifolds in a 6D manifold.
problem Counting special Lagrangian submanifolds in higher dimensions.
method Proving transversality for the moduli space of perturbed special Lagrangian submanifolds using a Lagrange multipliers problem.
result The moduli space is generically a set of isolated points.
In this paper we describe the cohomogeneity one special Lagrangian 3-folds in the cotangent bundle of the 3-sphere, also known in the physics literature as a deformed conifold. Our main result gives a global foliation of the deformed conifold by T^2-invariant special Lagrangian 3-folds, where the generic leaf is topolo…
Special Lagrangian submanifolds emerge from K3 surface collapse.
problem Understanding special Lagrangian submanifolds in K3 surface collapse.
method Lifting affine lines to degenerating sequences of special Lagrangian submanifolds.
result Constructing special Lagrangian two-spheres connecting Taub-NUT bubbles.
New homogeneous special Lagrangian submanifolds discovered in nearly Kähler CP3.
problem Exploring special Lagrangian submanifolds in nearly Kähler CP3.
method Intrinsically and extrinsically using moving frame and moment-type maps.
result Classification of totally geodesic special Lagrangian submanifolds and homogeneity of special Lagrangians.
The abstract discusses special Lagrangians and their flow, proving conjectures and observing related phenomena.
problem Existence and long-time existence of special Lagrangian representatives and Lagrangian mean curvature flow.
method Gibbons-Hawking ansatz, circle-invariant hyperkaehler 4-manifolds, Calabi-Yau 2-folds, Thomas conjecture, Thomas-Yau conjecture.
result Proves versions of the Thomas conjecture and Thomas-Yau conjecture.
This paper studies special Lagrangian submanifolds and their deformations.
problem Understanding the deformations of special Lagrangian submanifolds.
method Constructing a family of immersed special Lagrangian submanifolds as branched coverings.
result The existence of nondegenerate Z2 harmonic 1-forms is constrained. Butscher, D. Lee, Y. Lee, and Joyce constructed a special Lagrangian submanifold by gluing a Lawlor neck into a transverse intersection point of two special Lagrangian submanifolds. We prove a uniqueness theorem for the gluing of flat special Lagrangian tori of real dimension 3 in a flat complex torus of complex dimens…
Estimates for special Lagrangian curvature equations in critical and convex cases.
problem Interior estimates for special Lagrangian curvature equations.
method Establishes a priori interior curvature and gradient estimates.
result Proves interior curvature and gradient estimates for special Lagrangian curvature equations.
Paper proves uniqueness of special Lagrangian pair in Calabi-Yau 3-fold.
problem Existence and uniqueness of special Lagrangian pair of pants in Calabi-Yau 3-fold.
method Proves uniqueness of a special Lagrangian pair of pants with three asymptotically cylindrical ends.
result No other special Lagrangian pair satisfies the conjecture.
Construct special Lagrangian submanifolds in complex projective space.
problem Finding special Lagrangian submanifolds in complex projective space.
method Moment map technique and classification of cohomogeneity one actions.
result Examples of special Lagrangian submanifolds constructed.
Proves conditions for nearby special Lagrangians in Calabi-Yau manifolds.
problem Conditions for nearby special Lagrangians in Calabi-Yau manifolds.
method Analyzes fundamental group, Floer cohomology, and special Lagrangian conditions.
result Conditions for nearby special Lagrangians in Calabi-Yau manifolds.
Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.
problem Understanding geodesics in spaces of positive Lagrangian submanifolds.
method Cylindrical transform of geodesics, solving elliptic PDEs.
result Geodesics in positive Lagrangian spaces correspond to one-parameter families of special Lagrangian cylinders.
This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students i…
This is the second in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The first paper was math.DG/0008021, which studied special Lagrangian m-folds with large symmetry groups. The third is math.DG/0010036, which uses ideas from this paper to construct families of special Lag…
Method constructs special Lagrangian submanifolds using symmetries.
problem Constructing special Lagrangian submanifolds in Calabi-Yau manifolds.
method Using generalized perpendicular symmetries and moment maps of Lie group actions.
result Constructs non-flat Calabi-Yau manifolds with Stenzel metrics.
Generalizes Thomas-Yau theorem for special and minimal Lagrangians.
problem Proving uniqueness of special Lagrangians and minimal Lagrangians.
method Hamiltonian perturbations using Imagi, Joyce, and Oliveira dos Santos method.
result Generalized uniqueness theorem for special and minimal Lagrangians.
Study decomposability of Lagrangian classes on K3 surfaces.
problem Decomposability of Lagrangian classes on K3 surfaces.
method Lattice theory and special Lagrangian submanifolds.
result Prove decomposability of Kähler classes on dense subsets of the Kähler cone.
We give a necessary and sufficient condition for a special Lagrangian submanifold in C^n constructed by Lawlor being also special Lagrangian in C^n with the Fubini-Study form.
Proves smoothness and estimates for special Lagrangian solutions with semi-convexity.
problem Smoothness and estimates for special Lagrangian solutions.
method Viscosity solutions, smoothness, interior derivative estimates, sharpness of conditions.
result New Liouville theorem and effective Hessian estimates for special Lagrangian solutions.
Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.
problem Proving uniqueness of cylindrical tangent cones for special Lagrangians.
method Analyzing exact special Lagrangian submanifolds with multiplicity one and cylindrical tangent cones.
result The cylindrical tangent cones are unique under specific conditions.
We show the existence of special Lagrangian tori on one family of Borcea-Voisin threefolds. We also construct a family of special Lagrangian submanifolds on the total space of the canonical line bundle of projective spaces.
Study special Lagrangian submanifolds with edge singularities using elliptic theory.
problem Characterize the moduli space of deformations of special Lagrangian submanifolds with edge singularities.
method Use elliptic theory for edge-degenerate differential operators on singular manifolds.
result Obtain a general theorem describing the local structure of the moduli space.
This paper extends to dimension 4 the results in the article "Second Order Families of Special Lagrangian 3-folds" by Robert Bryant. We consider the problem of classifying the special Lagrangian 4-folds in C^4 whose fundamental cubic at each point has a nontrivial stabilizer in SO(4). Points on special Lagrangian 4-fol…
Constructs special Lagrangian submanifolds in Calabi-Yau 3-folds.
problem Constructing special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds.
method Constructs special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds fibered by K3 surfaces.
result Special Lagrangian submanifolds shrink to 1-dimensional graphs in the base as the 3-folds collapse.
Study on singularities in Lagrangian mean curvature flow with special Lagrangian cones.
problem Understanding singularities in Lagrangian mean curvature flow.
method Analysis of tangent flows and blowup limits of special Lagrangian cones.
result Uniqueness of tangent flows in dimension two and any dimension when the link is connected.
The study finds infinitely many special Lagrangian submanifolds in certain complex manifolds.
problem Existence of special Lagrangian submanifolds in log Calabi-Yau manifolds.
method Analyzes the existence of special Lagrangian submanifolds in log Calabi-Yau manifolds using Ricci-flat Kähler metrics.
result Proves existence of infinitely many disjoint special Lagrangian submanifolds in certain log Calabi-Yau manifolds.
In this paper, the relationship between the existence of special lagrangian submanifolds and the collapsing of Calabi-Yau manifolds is studied. First, special lagrangian fibrations are constructed on some regions of bounded curvature and sufficiently collapsed in Ricci-flat Calabi-Yau manifolds. Then, in the opposite d…
I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals. This method of constructing special Lagrangian submanifolds is well kn…
We construct smooth families of compact special Lagrangian submanifolds embedded in some toric hyper-K\"ahler manifolds, which never become holomorphic Lagrangian submanifolds via any hyper-Kähler rotations. These families converge to special Lagrangian immersions with self-intersection points in the sense of current…