New flows introduced for symplectic geometry.
problem No specific problem stated; focuses on new flows.
method Introduces several geometric flows on symplectic manifolds.
result Examples include the Hitchin gradient flow and dual Ricci flow.
Investigates metric degeneracies on symplectic leaves using a generalized gradient flow.
problem Degeneracies in metrics on symplectic leaves of Poisson manifolds.
method Introduces the generalized double bracket (GDB) vector field to generalize gradient dynamics.
result Identifies admissible regions where the double bracket metric remains non-degenerate on symplectic leaves, enabling GDB as a gradient flow.
Introduces a Morse complex on symplectic manifolds using gradient flows and proves its cohomology is independent of metrics and Morse functions.
problem Cohomology of symplectic manifolds under different metrics and Morse functions.
method Symplectic Morse complex with gradient flows and Witten deformation.
result Cohomology of the complex is isomorphic to Tsai, Tseng, and Yau's cohomology and independent of metrics and Morse functions.
This study proves the local existence of a symplectic gradient flow on a flat torus.
problem Proving the local existence of a symplectic gradient flow on a flat torus.
method Using a moment map and a DeTurck trick to make the flow strictly parabolic and showing local existence and regularity.
result The group of symplectomorphisms of the real four-dimensional torus is locally contractible.
Proposes a new Langevin flow approach for VAEs.
problem Difficulty in constructing low variance ELBO for VAEs with large datasets.
method Integrates Langevin dynamic with quasi-symplectic integrator to improve posterior estimation.
result Shows theoretical and practical effectiveness compared to gradient flow-based methods.
In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category.
We study the gradient flow lines of a Yang-Mills-type functional on the space of gauged holomorphic maps H(P,X), where P is a principal bundle on a Riemann surface Σ and X is a Kähler Hamiltonian G-manifold. For compact Σ, possibly with boundary, we prove long time existence of the gradient flow. …
Constructs a Morse-Bott function on symplectic Grassmannians.
problem Defines a function on symplectic Grassmannians.
method Uses a compatible linear complex structure to construct a quadratic Morse-Bott function.
result Critical loci consist of subspaces splitting into isotropic and complex parts.
The paper tackles isotropy of symplectic forms using Hodge flows.
problem Whether symplectic forms in a given class are isotropic.
method Introduces nonlinear Hodge heat flows to study isotropy.
result The flow converges to the symplectic form ω smoothly for any initial symplectic form in the class. Characterizes Anosov flows in 3D using symplectic and contact geometry.
problem Understanding Anosov flows in 3D.
method Purely contact and symplectic geometric methods.
result Characterization of Anosov flows based on Reeb flows and underlying (bi)-contact structures.
Ancient symplectic solutions to mean curvature flow are flat.
problem Understanding ancient solutions to mean curvature flow in symplectic geometry.
method Using a complex phase map to prove a Bernstein theorem.
result Ancient solutions to the symplectic mean curvature flow are flat.
Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
Extends symplectic flow results to foliations.
problem Applying hard Lefschetz theorem to symplectic foliations.
method Generalizes results from symplectic flows to foliations.
result Extends transversal hard Lefschetz theorem to transversely symplectic foliations.
The study finds static solutions in symplectic curvature flow in 4D.
problem Finding static solutions in symplectic curvature flow in 4D.
method Derived a local normal form for static solutions and used Cartan-Kahler theorem for solitons.
result Every complete static solution to symplectic curvature flow in 4D is Kahler-Einstein.
The paper studies Hamiltonian flows for pseudo-Anosov mapping classes on surfaces.
problem Understanding the dynamics of pseudo-Anosov mapping classes on Teichmüller spaces.
method Explicit formulae for Hamiltonian flows generated by invariant functions.
result Hamiltonian flows coincide with the action of pseudo-Anosov homeomorphisms at time one.
Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.
Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.
problem Preserving symplectic forms under deformations on Calabi-Yau manifolds.
method Dynamical stability of symplectic curvature flow.
result Any small symplectic deformation of a Kähler form remains Kähler on a compact Calabi-Yau manifold.
The Type IIA flow converges on symplectic manifolds, with singularity models identified.
problem Little was known about the singularities of the Type IIA flow.
method Formulated and proved convergence theorems for the Type IIA flow.
result Identified singularity models for the Type IIA flow.
Ancient solutions found for a specific flow on symplectic half-flat structures.
problem Existence of solutions for a particular geometric flow.
method Analyzes Type IIA flow on symplectic half-flat SU(3)-structures.
result Existence of ancient, immortal, and eternal solutions under suitable conditions.
New flow connects symplectic maps to hyperKähler geometry.
problem Understanding symplectic maps and their geometry.
method Established a correspondence between symplectic diffeomorphisms and hyperKähler moment maps.
result Introduced a new flow, the modified moment map flow.
This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [Y Eliashberg, A Givental and H Hofer, Introduction to Symplectic Field Theory, Geom. Funct. Anal. Special Volume, Part II (2000) 560--673]. We prove compactness results for moduli spaces of holomorphic curves arising in…
J. Streets and G. Tian recently introduced symplectic curvature flow, a geometric flow on almost Kähler manifolds generalising Kähler-Ricci flow. The present article gives examples of explicit solutions to this flow of non-Kähler structures on several nilmanifolds and on twistor fibrations over hyperbolic space studied…
Study of Type IIA flow on symplectic Lie algebras for geometric structures.
problem Detecting geometric structures like Lagrangian torus fibrations and harmonic almost complex structures.
method Investigate F-harmonic forms and the long-time behavior of the Type IIA flow. result The Type IIA flow helps in detecting desired geometric structures.
We introduce a parabolic flow of almost Kahler structures, providing an approach to constructing canonical geometric structures on symplectic manifolds. We exhibit this flow as one of a family of parabolic flows of almost Hermitian structures, generalizing our previous work on parabolic flows of Hermitian metrics. We e…
Starting from the vortex filament flow introduced in 1906 by Da Rios, there is a hierarchy of commuting geometric flows on space curves. The traditional approach relates those flows to the nonlinear Schrödinger hierarchy satisfied by the complex curvature function of the space curve. Rather than working with this infin…
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 …
Rabinowitz Floer homology is the semi-infinite dimensional Morse homology associated to the Rabinowitz action functional used in the pioneering work of Rabinowitz. Gradient flow lines are solutions of a vortex-like equation. In this survey article we describe the construction of Rabinowitz Floer homology and its applic…
We continue studying a parabolic flow of almost Kähler structures introduced by Streets and Tian which naturally extends Kähler-Ricci flow onto symplectic manifolds. In the system of primarily the symplectic form, almost complex structure, Chern torsion and Chern connection, we establish new formulas for the evolutions…
The study connects ECH capacities to Anosov flows, proving infinite capacities and obstructions.
problem Understanding ECH capacities and their relation to Anosov flows.
method Relating ECH capacities to Anosov flows dynamics, proving infinite capacities and obstructions.
result ECH capacities are infinite for many symplectic 4-manifolds, including cotangent disk bundles over surfaces of genus at least two.
Derivative estimates for pluriclosed flow control curvature and torsion.
problem Deriving derivative estimates for the pluriclosed flow.
method Control higher order derivatives of Chern curvature and torsion using Chern curvature; derive an estimate for torsion tensor using Chern Ricci curvature in dimension two; find a monotonic quantity in Hermitian-symplectic case.
result All Hermitian-symplectic solitons are Kähler Ricci solitons.
Let (M,g) be a Kähler surface, and Σ an immersed surface in M. The Kähler angle of Σ in M is introduced by Chern-Wolfson \cite{CW}. Let (M,g(t)) evolve along the Kähler-Ricci flow, and Σt in (M,g(t)) evolve along the mean curvature flow. We show that the Kähler angle $α…
In this paper, we construct finite blow-up examples for symplectic mean curvature flows and we study properties of symplectic translating solitons. We prove that, the Kähler angle α of a symplectic translating soliton with max∣A∣=1 satisfies that sup∣α∣>4π∣T∣+1∣T∣ where T is the direction in…
The paper studies a degenerate equation related to Kähler-Ricci flow on symplectic quotients.
problem Finite time singularities of the Kähler-Ricci flow on symplectic quotients.
method Interpreting the V-soliton equation and reducing it to a scalar equation on Kähler potentials. result Preliminary estimates for the scalar equation on compact Kähler manifolds.
We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for stu…
Study of Kähler-Ricci flow on toric Fano varieties with preserved symplectic condition.
problem Analyzing the generalized Kähler-Ricci flow on toric Fano varieties.
method Establishing global existence, deriving entropy and energy functionals.
result Convergence of nonsingular solutions at infinity and weak convergence of the flow.
Stability of Type IIA flow ensures Kähler properties of Calabi-Yau 3-folds.
problem Ensuring the Kähler property of Calabi-Yau 3-folds under symplectic deformations.
method Established dynamical stability of Type IIA flow near stationary points.
result Stability of Type IIA flow ensures the stability of Kähler properties under symplectic deformations.
Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to ζ-flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of ζ-flat bundles. result Novel geometric flows and characteristic classes of ζ-flat bundles are described. Optimizes functions on Lie groups using generalized eigenvalue problems.
problem Optimization on Lie groups with specific applications to eigenvalue problems.
method Generalizes NAG principle to Lie groups, resulting in continuous Lie-NAG dynamics converging to local optima.
result Discretized Lie-NAG dynamics yield structure-preserving optimization algorithms with faithful energy behavior.
New method solves optimization problems on manifolds using symplectic integrators.
problem Optimization tasks on manifolds with nonlinear constraints.
method Dissipative extension of Dirac's theory of constrained Hamiltonian systems and geometric/symplectic numerical integrators.
result Developed algorithms achieve optimal convergence rates locally.
We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,Ω) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all suffici…
Geometric flow on symplectic manifolds connects to Type IIA string theory.
problem Finding optimal almost-complex structures compatible with symplectic forms.
method Introduced a geometric flow on 6-dimensional symplectic manifolds with SU(3) holonomy.
result The flow leads to Ricci-flat Kähler metrics and optimal almost-complex structures.
This is an exposition of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. The original work appeared in [1].
For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a nondegenerate (i.e. symplectic) magnetic field has periodic orbits on a sequence o…
In this paper we mainly study the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K ähler-Einstein surface. We show the relation between the maximum of the Kähler angle and the maximum of ∣H∣2 on the limit flow.
New flow category for contact manifolds from Reeb orbits.
problem No direct problem stated; focuses on new construction.
method Adapting Kuranishi charts to contact setting, associating flow category based on Reeb orbits and pseudo-holomorphic buildings.
result Lifts contact homology and associates flow bimodule to exact symplectic cobordisms.
Invites study of contact structures and Reeb flows dynamics.
problem Relating dynamics of two Reeb flows of the same contact structure.
method Gathers results and poses many questions and conjectures.
result Many new questions and conjectures posed.
Excises interesting subsets from symplectic manifolds.
problem Excision of interesting closed subsets from symplectic manifolds.
method Time-independent incomplete Hamiltonian flows.
result Generalizes a result about excision of a ray.
Let (M,gˉ) be a Kähler surface with a constant holomorphic sectional curvature k>0, and Σ an immersed symplectic surface in M. Suppose Σ evolves along the mean curvature flow in M. In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve i…