The paper proves stability of translating solitons in Lagrangian geometry.
problem Stability of translating solitons in Lagrangian geometry.
method Proof of Lagrangian -stability.
result Lagrangian translating solitons are Lagrangian -stable.
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48 results for “Lagrangian geometry”
The paper explores the geometry of Lagrangian Grassmannians and their connection to PDEs.
problem Understanding the geometric properties of Lagrangian subspaces.
method Thorough review of geometric properties and their relation to PDEs.
result Hypersurfaces in the Lagrangian Grassmannian correspond to second-order PDEs.
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 paper explores similarities in even and odd-dimensional geometry.
problem Understanding the Lagrangian Grassmannian in cosymplectic geometry.
method Study of compatible co-complex structures, Moser's trick, and Weinstein 1-form derivation.
result The de Rham class of the Weinstein 1-form is a co-flux.
Study canonical curves and Kropina metrics in Lagrangian contact geometry.
problem Characterize canonical curves and their relationship to Lagrangian contact structures.
method Construct Fefferman-type spaces, analyze chains and null-chains, use Kropina metrics, apply Fermat principle.
result Chains and null-chains in integrable Lagrangian contact structures are geodesics of Kropina metrics.
The paper explores Hamiltonian stationary Lagrangian fibrations in various geometries.
problem Understanding HSLAG submanifolds and fibrations in different geometries.
method Deforming toric Kähler metrics into non-toric almost Kähler metrics to find HSLAG submanifolds.
result A large class of Hamiltonian stationary Lagrangian fibrations are found.
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.
Explains examples of Lagrangian flow with circle symmetry.
problem Understanding Lagrangian flow with symmetry.
method Examining specific examples in .
result Shows various types of flow, including compact and non-compact.
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…
Geoffrey Martin's theorem proves normal forms for Lagrangian submanifolds in multisymplectic geometry.
problem Normal forms for Lagrangian submanifolds in multisymplectic geometry.
method Detailed, self-contained proof of normal form theorem, including necessary results in foliated differential topology.
result Geoffrey Martin's theorem provides a normal form for Lagrangian submanifolds in multisymplectic geometry.
Study of Lagrangian submanifolds in para-complex Euclidean space.
problem Characterizing and understanding Lagrangian submanifolds in para-complex Euclidean space.
method Analyzing curvature equations and extrinsic geometry of submanifolds.
result Characterization of minimal Lagrangian surfaces and self-similar solutions of Mean Curvature Flow.
Proves Hölder-type inequality for Lagrangians' distance.
problem Understanding the symplectic geometry of Lagrangians.
method Developed methods from previous works to establish the inequality.
result Established a Hölder-type inequality for the Hausdorff distance between Lagrangians.
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…
Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper we study a generalization, which consists of replacing th…
It is known that some equations of differential geometry are derived from variational principle in form of Euler-Lagrange equations. The equations of geodesic flow in Riemannian geometry is an example. Conversely, having Lagrangian dynamical system in a manifold, one can consider it as geometric equipment of this manif…
A new Lagrangian formulation of the Raychaudhuri equation in non-Riemannian geometry.
problem Formulating the Raychaudhuri equation in non-Riemannian geometries.
method Established a formal connection between the expansion scalar and the cross-sectional volume of the congruence. Derived a Lagrangian and Hamiltonian formulation.
result The expansion scalar equals the fractional rate of change of volume, weighted by a scalar factor.
The paper proves a version of the Arnol'd conjecture for Lagrangian submanifolds in conformal symplectic geometry.
problem Understanding the behavior of Lagrangian submanifolds in conformal symplectic geometry.
method Proves a version of the Arnol'd conjecture for Lagrangian submanifolds in conformal symplectic manifolds.
result The number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of the Lee form for generic isotopies.
Study shows how to preserve Lagrangian condition in mean curvature flow on Kim-McCann metrics.
problem Preserving Lagrangian condition in mean curvature flow on Kim-McCann metrics.
method Expressed mean curvature flow within generalized mean curvature flow framework.
result Lagrangian condition is preserved along the flow.
Book teaches how Lagrangian torus fibration base geometry can be read off.
problem Understanding geometry of Lagrangian torus fibrations.
method Integral affine structure on fibration base for total space geometry.
result Read off interesting geometry of total space from base.
New approach to Lagrangian systems using intrinsic geometry.
problem Developing a new framework for Lagrangian systems.
method Direct reformulation of Hamiltonian formalism, introduction of spatial equation and spatial-gauge symmetry.
result Covariant and non-covariant canonical variational principles demonstrated for Maxwell equations.
Study shows special Kähler geometry on base of holomorphic Lagrangian fibrations implies projective space.
problem Understanding the geometry of holomorphic Lagrangian fibrations.
method Using special Kähler geometry and parallel splitting of tangent bundles.
result Holomorphic Lagrangian fibrations over non-projective spaces are impossible.
Study of Lagrangian submanifolds with Riemannian bounds and their metric properties.
problem Understanding the geometry and topology of Lagrangian submanifolds with Riemannian constraints.
method Investigation of metric properties and symplectic structures on spaces of Lagrangian submanifolds with uniform Riemannian bounds.
result There are at most countably many Hamiltonian isotopy classes of exact Lagrangian submanifolds in a Liouville manifold.
The paper proves finiteness for stable Lagrangian fibrations with a given divisor.
problem Finiteness of deformation classes of hyperkähler Lagrangian fibrations.
method Survey and proof of finiteness for stable Lagrangian fibrations with a given discriminant divisor.
result Finiteness for stable Lagrangian fibrations with a given discriminant divisor.
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riema…
Researchers generalize cosmological models using Finsler geometry.
problem Cosmological models that closely approximate pseudo-Riemannian geometry.
method Identifying Lie Algebra of symmetry generators for spatially homogeneous and isotropic Finsler geometries.
result Found the most general spatially homogeneous and isotropic Berwald spacetimes.
Construct special Lagrangian fibrations on abelian varieties using retraction techniques.
problem Constructing special Lagrangian fibrations on abelian varieties.
method Explicit construction using special techniques in non-Archimedean geometry.
result Solved a conjecture of Kontsevich-Soibelman for finite quotients of abelian varieties.
This Ph.D. thesis is devoted to the constructions of Lagrangian formulation on Finsler and Kawaguchi manifolds. While Finsler geometry is a natural extension of Riemannian geometry, Kawaguchi geometry is the extension of Finsler geometry to higher order derivatives and to k-dimensional parameter space. The latter exten…
Internal Lagrangians derived from variational principles.
problem Reproducing the principle of stationary action in variational geometry.
method Introducing stationary points of internal Lagrangians, establishing connections with symmetries and conservation laws, and investigating relations between non-degenerate and internal Lagrangians.
result Noether's theorem reformulated in terms of internal Lagrangians.
We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the…
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 paper studies special Lagrangian manifolds using algebraic topology.
problem Understanding special Lagrangian submanifolds in calibrated geometries.
method Algebraic topology of Grassmannian spaces, focusing on cohomology rings.
result Various results on the topology of Grassmannian spaces and special Lagrangian embeddings.
New approach solves problems in gauge theory and symplectic geometry.
problem Instanton Floer homology and Lagrangian Floer theory.
method Combination of cobordism type argument and homological algebra.
result Various problems solved and proofs simplified.
Study SYZ transforms for immersed Lagrangian multi-sections in symplectic geometry.
problem Understanding the geometry of SYZ transforms on Lagrangian torus fibrations.
method Investigation of Lagrangian surgery and extension of holomorphic vector bundles via SYZ transform for immersed Lagrangian multi-sections.
result New equivalence in the immersed Fukaya category invariant under immersed Floer cohomology, providing a mirror to isomorphism of holomorphic vector bundles.
We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the questi…
Introduces internal Lagrangians for differential equations and connects them to presymplectic structures.
problem Understanding the geometry of differential equations and their solutions.
method Develops a spectral sequence related to internal Lagrangians and investigates connections to presymplectic structures.
result Interprets a term in Vinogradov's spectral sequence for gauge theories.
Lagrangian branes with boundary and symplectic methods for stable generalized complex manifoldsmath.DG
This paper introduces new Lagrangian branes in stable generalized complex manifolds.
problem Understanding stable generalized complex manifolds and their properties.
method Using log symplectic geometry and Floer theory techniques.
result Lagrangian branes with boundary are introduced and their properties are studied.
Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.
problem Understanding the structure of holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds.
method Analyzing the properties of Lagrangian fibrations and special Kähler structures.
result Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.
The aim of the present text is twofold: to provide a compendium of Lagrangian and Hamiltonian geometries and to introduce and investigate new analytical Mechanics: Finslerian, Lagrangian and Hamiltonian. The fundamental equations (or evolution equations) of these Mechanics are derived from the variational calculus appl…
We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant.
We discuss the interplay between lagrangian distributions and connections in symplectic geometry, beginning with the traditional case of symplectic manifolds and then passing to the more general context of poly- and multisymplectic structures on fiber bundles, which is relevant for the covariant hamiltonian formulation…
This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.
problem Understanding the Hamiltonian stationarity of twisted Lagrangian tori in C^2.
method Investigation of differential geometry of twisted tori, including product and Chekanov's exotic tori.
result Only product tori are minimal under Hamiltonian deformations, indicating Chekanov's exotic tori are not area minimal.
The paper studies Weyl structures on parabolic geometries and their properties.
problem Understanding Weyl structures on parabolic geometries.
method Analyzes a natural affine bundle and its sections, showing connections to reductive Cartan geometries and bi-Lagrangian structures.
result Weyl structures on torsion-free parabolic geometries are Einstein with non-zero scalar curvature.
Abstract mathematical formulas for statistical structures and curvatures.
problem Developing formulas for statistical structures and curvatures.
method Proving new formulas and theorems for statistical structures and curvatures.
result Generalized formulas for statistical structures and curvatures.
Reducing almost Lagrangian structures and almost CR geometries to partially integrable structuresmath.DG
This paper demostrates a method for analysing almost CR geometries , by uniquley defining a partially integrable structure from the same data. Thus two almost CR geometries and are equivalent if and and only if they generate isomorphic induced partially integrable CR geometries …
Geometric heat-flow theory detects Lagrangian coherent structures.
problem Detecting Lagrangian coherent structures in advective-diffusive systems.
method Transform Eulerian advection-diffusion to Lagrangian coordinates, then solve the resulting geometric heat equation.
result LCSs are boundaries of metastable sets under Lagrangian diffusion.
The author exposes the metrical multi-time Lagrange geometry of physical fields which naturally generalizes the classical Lagrangian developped by Miron and Anastasiei. In other words, one constructs a natural theory of physical fields on the 1-jet fibre bundle, attached to a Kronecker h-regular multi-time Lagrangian w…
The paper studies deformations of Lagrangian submanifolds using algebraic tools.
problem Deformation theory of Lagrangian submanifolds in symplectic geometry.
method Graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem, attaching an -algebra to each submanifold.
result Controls the deformation theory of Lagrangian -submanifolds using an -algebra.
Geometrizes second order Lagrangians transformations.
problem No specific problem stated; focuses on geometrization.
method Building a proper Tulczyjew's triplet.
result Symplectic relation between Ostrogradsky-Legendre and Schmidt-Legendre transformations.