Minimal Lagrangian connections on surfaces classified.
problem Classifying minimal Lagrangian connections on compact surfaces.
method Introduced and classified minimal Lagrangian connections on compact surfaces.
result Every properly convex projective surface arises from a unique minimal Lagrangian connection.
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…
Develops a new method for minimal Lagrangian surfaces in complex quadrics.
problem Finding minimal Lagrangian surfaces in complex quadrics.
method Loop group representation and flat connections.
result Equivalence of minimality and flatness of connections, explicit families of examples.
Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of Riemannian minimality. A Lagrangian submanifold is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero. This notion was introduced in the work of Y.-G. Oh in connection with the celebr…
We study singularities of Lagrangian mean curvature flow in $\C^n$ when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonia…
Minimal Lagrangian submanifolds are equivalent to their reduced forms in Kähler quotients.
problem Finding minimal Lagrangian submanifolds in complex manifolds.
method Using Kähler reductions and Hsiang-Lawson metrics to relate minimality of submanifolds to their reduced forms.
result The minimality of a Lagrangian submanifold is equivalent to that of its reduced form in a Kähler quotient.
Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.
problem Characterizing and constructing minimal Lagrangian surfaces in complex hyperbolic quadric.
method Loop of flat connections, isometric deformations, DPW-type representation.
result Explicit examples of minimal Lagrangian surfaces, including catenoid-type examples.
We study the formation of singularities for the mean curvature flow of monotone Lagrangians in $\C^n$. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union of area-minimizing Lagrangian cones (Slag cones). When n=2, we can improve this…
Embeds complex 3-manifolds into symplectic space.
problem Embedding non-orientable 3-manifolds into symplectic spaces.
method Lagrangian embedding of specific 3-manifolds into standard symplectic 6-space.
result Minimal Maslov number of embedding is 1.
In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP^2#3(-CP^2) which contains a sym…
The paper establishes a connection between superminimal surfaces and Lagrangian submanifolds in twistor spaces.
problem Understanding the geometric relationship between superminimal surfaces and Lagrangian submanifolds in twistor spaces.
method Proves a bijective correspondence between superminimal surfaces and Lagrangian submanifolds of twistor spaces, using specific constructions and projections.
result Produces many Lagrangian submanifolds of twistor spaces that are also minimal.
Global minimizers exist for Tonelli Lagrangians on half-Lie groups.
problem Existence and properties of minimizers for Lagrangians on infinite-dimensional spaces.
method Introduced Tonelli Lagrangians on half-Lie groups, proved existence of minimizers and flow lines.
result Global minimizers exist above certain energy thresholds.
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.
Study on exact Lagrangian submanifolds in unit ball with Legendrian boundary.
problem Exact Lagrangian submanifolds with Legendrian boundary in unit ball.
method Uses Liouville form and boundary unique continuation for differential forms.
result Equatorial n-disk rigidity for compact exact Lagrangian self-similar submanifolds. 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 Σs∞ at a singular point (X0,T0) of a symplectic mean curvature flow Σt or of a Lagrangian mean curvature flow Σt is …
H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with nega…
Minimal Lagrangian submanifolds deform to J-minimal ones under small perturbations.
problem Understanding how minimal Lagrangian submanifolds behave under small perturbations in Kaehler-Einstein manifolds.
method Analyzing the deformation of minimal Lagrangian submanifolds under Kaehler-Einstein perturbations.
result Deformed submanifolds remain J-minimal under certain conditions.
Construct minimal Lagrangian surfaces in complex projective plane via loop group method.
problem Construct minimal Lagrangian immersions from arbitrary Riemann surfaces into complex projective plane.
method Loop group method, perturbed equivariant minimal Lagrangian surfaces, Delaunay cylinders approximation.
result Construct a class of minimal Lagrangian cylinders approximating Delaunay cylinders.
Survey on real forms of a complex equation and their connection to surface theory.
problem Describing real forms of the complex A2(2)-Toda equation and their geometric implications. method Analyzing the integrability of Maurer-Cartan forms for different real forms of loop groups.
result Each real form of A2(2) corresponds to a specific surface class with integrable frames. 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.
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…
Minimal Lagrangians in certain curved spaces are stable under specific flows.
problem Stability of minimal Lagrangians in Kähler-Einstein manifolds of non-positive curvature.
method Proved stability under Lagrangian mean curvature flow.
result Equivalence between linear and dynamical stability for C1-close Lagrangians. Let N be a complete, simply-connected surface of constant curvature κ\leq 0. Moreover, suppose that Ωand \tildeΩ are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from Ωto \tildeΩ whose graph is a minimal submanifold of N \times N.
Study minimal Lagrangian surfaces in complex projective plane, focusing on contractible cases.
problem Construct minimal Lagrangian surfaces in complex projective plane.
method Loop group method
result Presented new classes of minimal Lagrangian surfaces.
Study confirms conjecture: minimal Lagrangian surfaces with Legendrian boundary are rigid.
problem Characterize minimal Lagrangian surfaces with Legendrian capillary boundary.
method Analyzes surfaces in B4 with Legendrian boundary on S3. result Equatorial plane disks and catenoids are the only minimal Lagrangian surfaces with Legendrian capillary boundary.
Study complex properties of minimal Lagrangian submanifolds in Kaehler spaces.
problem Complex properties of minimal Lagrangian submanifolds in Kaehler spaces.
method Mix of holomorphic curve techniques and convexity results.
result Minimal Lagrangians do not admit fillings by holomorphic discs in negative curvature case.
Study f-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials.
problem Variational properties of f-minimal Lagrangian submanifolds. method Derive second variation formula and study stability of solitons.
result Stability of expanding and translating solitons for LMCF.
In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomegeneity one. We characterize them as the only minimal Lagrangian submanifolds in CH^n foliated by umbilical hypersurfaces of Lagrangian s…
Derives Lagrangian for minimal surfaces, proving tangential variations vanish.
problem Variational calculus for minimal surfaces.
method Lagrangian formulation, pullback covariant derivative, geometric argument.
result Tangential variations vanish for minimal surfaces.
Study Lagrangian submanifolds from minimal surfaces in S³.
problem Characterize Lagrangian submanifolds of nearly Kähler S³ × S³.
method Relate to minimal surfaces in S³ and use angle functions.
result Construct a family of Lagrangian immersions from minimal surfaces.
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.
The paper characterizes minimal surfaces and Lagrangian surfaces in complex projective space.
problem Characterizing minimal surfaces and Lagrangian surfaces in complex projective space.
method Using Ruh-Vilms type theorems.
result Characterizes minimal surfaces and Lagrangian surfaces in complex projective space.
Maps Lagrangian submanifolds to quaternionic projective space, proving one-to-one correspondences.
problem Constructing maps between Lagrangian submanifolds and quaternionic projective spaces.
method Explicit construction of maps from minimal δ(2)-ideal Lagrangian submanifolds of Cn to HPn−1. result One-to-one correspondences between minimal Lagrangian surfaces in CP2 and minimal totally complex surfaces in HP2. Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.
problem Counting minimal Lagrangians in hyperbolic surfaces.
method Uses Mirzakhani functions to show growth rate and proves rigidity of area spectrum.
result Number of minimal Lagrangians grows as A6(k−1). Study minimal Lagrangian submanifolds of complex hyperquadric.
problem Classify minimal Lagrangian submanifolds of complex hyperquadric.
method Use non-integrable almost product structures and local angle functions.
result Classify minimal Lagrangian submanifolds with constant sectional curvatures and those with coinciding local angle functions.
We give a survey of various existence results for minimal Lagrangian graphs. We also discuss the mean curvature flow for Lagrangian graphs.
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 studies timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.
problem Characterizing timelike minimal Lagrangian surfaces in indefinite complex hyperbolic space.
method Defining natural Gauss maps and proving a Ruh-Vilms type theorem.
result Timelike minimal Lagrangian surfaces correspond to the fifth real form of the complex affine Kac-Moody algebra.
The paper classifies product minimal Lagrangian submanifolds in complex space forms.
problem Understanding minimal Lagrangian submanifolds in complex space forms.
method Examining submanifolds as Riemannian products with constant sectional curvature.
result Complete classification of product minimal Lagrangian submanifolds.
Given a complex structure J on a real (finite or infinite dimensional) Hilbert space H, we study the geometry of the Lagrangian Grassmannian Λ(H) of H, i.e. the set of closed linear subspaces L⊂H such that J(L)=L⊥. The complex unitary group U(HJ), consisting of the elements of the orthogona…
Study minimal Lagrangian tori on Kähler manifolds, answering questions about their existence and stability.
problem Characterize minimal Lagrangian tori on Kähler manifolds.
method Investigate orbits of torus actions, analyze stability, and relate to ambient geometry.
result Partial answers to questions about minimal Lagrangian tori existence and stability.
Most Lagrangian torus orbits in complex projective space are not volume minimizing.
problem Identifying volume minimizing Lagrangian orbits in complex projective spaces.
method Analyzing Hamiltonian isotopies and volume minimization properties of Lagrangian torus orbits.
result Almost all Lagrangian torus orbits are not Hamiltonian volume minimizing.
We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized.
The paper connects Weil-Petersson homeomorphisms to maximal surfaces in anti-de Sitter space.
problem Understanding Weil-Petersson homeomorphisms in Teichmüller theory.
method Analyzing maximal spacelike surfaces in anti-de Sitter space and their asymptotic boundaries.
result A homeomorphism is Weil-Petersson if and only if its minimal lagrangian extension has square-integrable Beltrami differential.
The paper discusses new Lagrangian constructions and examples.
problem Exploring new Lagrangian constructions in complex projective spaces.
method Generalized Delaunay construction among minimal Lagrangians.
result Uncountably many new special Lagrangian cones found.
Paper solves flat bi-Lagrangian structure problems in ray space.
problem Existence of flat bi-Lagrangian structures in ray space.
method Established geometric conditions for flat canonical connections.
result Complete solutions to two problems regarding flat bi-Lagrangian structures.
Geodesic connects Lagrangian graphs over torus in complex space.
problem Existence of geodesic connecting Lagrangian graphs in complex space.
method Formulated as a degenerate elliptic equation, solved via Dirichlet problem.
result Geodesic connecting Lagrangian graphs can be constructed.
Lower bound found for energy on specific Lagrangian tori in complex projective space.
problem Finding a lower bound for the energy functional on Lagrangian tori in CP2. method Analyzing the energy functional on a family of Hamiltonian minimal Lagrangian tori.
result Proved that the energy of certain Hamiltonian minimal Lagrangian tori is strictly larger than the Clifford torus.