arXiv research

Ancient symplectic solutions to mean curvature flow are flat.

The study finds static solutions in symplectic curvature flow in 4D.

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.

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 $Σ_s^\infty$ at a singular point $(X_0, T_0)$ of a symplectic mean curvature flow $Σ_t$ or of a Lagrangian mean curvature flow $Σ_t$ is …

Derivative estimates for pluriclosed flow control curvature and torsion.

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…

Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.

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…

Let $(M,\overline{g})$ 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 $(M,\overline{g}(t))$ evolve along the Kähler-Ricci flow, and $Σ_t$ in $(M,\overline{g}(t))$ 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 $\sup |α|>\fracπ{4}\frac{|T|}{|T|+1}$ where $T$ is the direction in…

Let $(M,\bar{g})$ 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…

Let Σbe a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold M. We consider the evolution of Σin the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel Kähler forms…

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. First, we prove that if the ratio $λ$ of the maximum and the minimum of the holomorphic sectional curvatures $< 2$, then there exists a positive constant $δ>\frac{29(λ-1)}{\sqrt{(48-24λ)^{2}+(29λ-29…

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.

Explains examples of Lagrangian flow with circle symmetry.

We prove that for a mean curvature flow of a compact symplectic surface in a compact Kaehler-Einstein surface, the tangent cone at the first blow-up time consists of a finite union of more than two 2-planes in $R^4$ which are complex in a complex structure on $R^4$.

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…

Recently, J. Streets and G. Tian introduced a natural way to evolve an almost-Kähler manifold called the symplectic curvature flow, in which the metric, the symplectic structure and the almost-complex structure are all evolving. We study in this paper different aspects of the flow on locally homogeneous manifolds, incl…

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2, then there exists a positive constant $δ$ depending on the ratio such that $\cosα\ge…

New geometric flow called hypersymplectic flow studied, proving key properties.

Let $M$ be a Kähler-Einstein surface with positive scalar curvature. If the initial surface is sufficiently close to a holomorphic curve, we show that the mean curvature flow has a global solution and it converges to a holomorphic curve.

An almost Kähler structure on a symplectic manifold $(N, ω)$ consists of a Riemannian metric $g$ and an almost complex structure $J$ such that the symplectic form $ω$ satisfies $ω(\cdot, \cdot)=g(J(\cdot), \cdot)$. Any symplectic manifold admits an almost Kähler structure and we refer to $(N, ω, g, J)$ as an almost Käh…

We apply the mean curvature flow to deform symplectomorphisms of $\mathbb{CP}^n$. In particular, we prove that, for each dimension n, there exists a constant $Λ$, explicitly computable, such that any $Λ$-pinched symplectomorphism of $\mathbb{CP}^n$ is symplectically isotopic to a biholomorphic isometry.

The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity, global existence and convergence of the flow in various ambient spaces and codimensio…

The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrange distribution) on the symplectic manifold. It is shown that the negativity of the reduced curvature implies the hyperbolicity of any compact invariant set of the Hamiltonian flow restricted to a prescribed…

New flows introduced for symplectic geometry.

We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety …

We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kähler case, by its analogues. To this end we…

Study rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.

Study geometric mKdV flows for Legendrian curves in a 3-sphere.

The paper tackles isotropy of symplectic forms using Hodge flows.

We prove some basic properties of Donaldson's flow of surfaces in a hyperkahler 4-manifold. When the initial submanifold is symplectic with respect to one Kähler form and Lagrangian with respect to another, we show that certain kinds of singularities cannot form, and we prove a convergence result under a condition rela…

Characterizes Anosov flows in 3D using symplectic and contact geometry.

Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.

We construct integrable hierarchies of flows for curves in centroaffine ${\mathbb R}^3$ through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and …

The aim of this paper is to study self-similar solutions to the symplectic cuvature flow on 6-dimensional nilmanifolds. For this purpose, we focus our attention in the family of symplectic Two- and Three-step nilpotent Lie algebras admitting a "minimal compatible metric" and we give a complete classification of these a…

Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of $2n$-dimensional nondegenerate hypersurfaces ruled by $n$-planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with r…

The Almost Hermitian Curvature flow was introduced by Streets and Tian in order to study almost hermitian structures, with a particular interest in symplectic structures. This flow is given by a diffusion-reaction equation. Hence it is natural to ask the following: which almost hermitian structures are dynamically stab…

Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and $\int_{\partial P} u ~ d σ< C_1,$ then there exists a constant $C_2$ depending only on $C_1$ and $P$ such that $\max_P u < C_2$. As an application, we sh…

In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit …

The paper connects moment maps to the stability of holomorphic fibrations.

We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233--245) to the general transversely symplectic foliatio…

The Type IIA flow converges on symplectic manifolds, with singularity models identified.

Ancient solutions found for a specific flow on symplectic half-flat structures.

New flow connects symplectic maps to hyperKähler geometry.

Study of Type IIA flow on symplectic Lie algebras for geometric structures.

Study Kähler-Ricci flow on rational homogeneous varieties using algebraic geometry and representation theory.

A hypersymplectic structure on a 4-manifold $X$ is a triple $\underlineω$ of symplectic forms which at every point span a maximal positive-definite subspace of $Λ^2$ for the wedge product. This article is motivated by a conjecture of Donaldson: when $X$ is compact $\underlineω$ can be deformed through cohomologous hype…