We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.
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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.
Paper proves stability of flow in cotangent bundle for special Lagrangian submanifolds.
problem Stability of generalized Lagrangian mean curvature flow in cotangent bundle.
method New derivative estimates to weaken initial conditions and remove curvature constraints.
result Stability of flow near special Lagrangian submanifolds in cotangent bundle.
Continues study on special Lagrangian graphs and flow solutions.
problem Long time existence and convergence of a class of fully nonlinear flows.
method Analyzes special Lagrangian graphs with prescribed second boundary conditions.
result Long time existence and convergence for a family of special Lagrangian graphs.
We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded…
The paper proves a Liouville theorem for special Lagrangian equations with convexity conditions.
problem Proving Liouville theorems for special Lagrangian equations with specific conditions.
method Using Neumann-Poincaré inequality, mean value inequality for superharmonic functions, and geometric measure theory.
result Derives global and interior Hessian estimates for solutions of special Lagrangian equations.
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.
In this paper, we prove some Bernstein type results for -dimensional minimal Lagrangian graphs in quaternion Euclidean space . In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in
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.
We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to Monge-Ampere Equations.
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.
The paper proves optimal smoothness for certain Lagrangian graphs with specific Hölder continuity.
problem Optimal regularity for Hölder continuous Hamiltonian stationary Lagrangian graphs.
method Establishing smoothness conditions based on Hölder exponent and Lagrangian phase properties.
result Smoothness of graphs is achieved when Hölder exponent is strictly greater than 1/3 and Lagrangian phase is supercritical.
The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
problem Mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.
method Analyzes the parabolic equation and Monge-Ampère type equation, proving smooth solutions and convergence to self-expanding solutions.
result Smooth solutions for specific nonlinear equations and convergence to self-expanding solutions.
Every finite graph arises as the singular set of a compact -d calibrated area minimizing surfacemath.DG
Every graph can be represented as a singular set of a special surface.
problem Representing any finite graph as the singular set of a compact 3D surface.
method Constructing a calibrated 3-dimensional homologically area minimizing surface with a special Lagrangian form.
result The singular set of the surface is precisely the given graph.
Study on evolving graphs of functions under mean curvature flow in R^n.
problem Proving long-time existence and convergence of special Lagrangian evolution equation.
method Consider the graph of a function on , deform it by mean curvature flow, and analyze under 2-positivity assumption.
result Proves long-time existence and convergence results under 2-positivity assumption, improving previous results.
This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in Existence of geodesics in the space of positive Lagrangians is an important step in…
Develops a method to construct entire minimal graphs of odd dimensions.
problem Constructing entire minimal graphs of odd dimensions and arbitrary codimensions.
method Evolving-plane ansatz reducing minimal surface system to geodesic equation on Grassmannian.
result Yields a rich family of explicit entire minimal graphs of odd dimension and arbitrary codimension.
The paper solves pseudo-convexity for special Lagrangian equations, with applications in mirror symmetry.
problem Existence of solutions to the Special Lagrangian Potential Equation.
method Explicitly computing boundary conditions for pseudo-convexity in potential theory.
result Interesting pseudo-convexity results for various related equations, including deformed Hermitian-Yang-Mills and curvature equations.
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.
Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problemmath.DG
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…
The article recovers the Smale conjecture on a Sasakian 3-sphere using Legendrian mean curvature flow.
problem Recovering the Smale conjecture on a Sasakian 3-sphere.
method Using Legendrian mean curvature flow to deform area-preserving contactomorphisms to isometries.
result Obtained the minimal Legendrian graph in S² × S³.
In this paper, a class of fully nonlinear flows with nonlinear Neumann type boundary condition is considered. This problem was solved partly by the first author under the assumption that the flow is the parabolic type special Lagrangian equation in . We show that the convexity is preserved for solution…
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.
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…
Special Lagrangian Conesmath.DG
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 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…
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 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.
Nearby Special Lagrangiansmath.SG
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.
We introduce a notion of vanishing Maslov index for lagrangian varifolds and lagrangian integral cycles in a Calabi-Yau manifold. We construct mass-decreasing flows of lagrangian varifolds and lagrangian cycles which satisfy this condition. The flow of cycles converges, at infinite time, to a sum of special lagrangian …