The symplectic Thom conjecture

Arnold-Thom 猜想关于曲面的到达时间

Arnold-Thom conjecture for the arrival time of surfaces

本文在具有颈缩或非退化圆柱奇异性的平均曲率流到达时间函数背景下，证明了 Łojasiewicz 定理和 Arnold 猜想，并推广至所有平均凸的曲面平均曲率流。

The Weinstein Conjecture in Product of Symplectic Manifolds

In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product $P_1\times P_2$ of two strongly geometrically bounded symplectic manifolds under some conditions with $P_1$. In particular, if $N$ is a closed manifold or a noncompact manifold of finite topological type, our result implies that the Weinstein conjecture in $\mathbb{C}\mathbb{P}^2\times T^*N$ holds.

非线性演化方程的Thom梯度猜想

Thom's gradient conjecture for nonlinear evolution equations

本文将Thom梯度猜想推广至无穷维问题并解决，基于已有研究聚焦黎曼流形上的非线性演化，覆盖多类几何PDEs，证实极限方向唯一性并刻画收敛速率与可能极限方向。

On the Lusternik-Schnirelmann category of symplectic manifolds and the Arnold conjecture

We prove that the Lusternik-Schnirelmann category $cat(M)$ of a closed symplectic manifold $(M, ω)$ equals the dimension $dim(M)$ provided that the symplectic cohomology class vanishes on the image of the Hurewicz homomorphism. This holds, in particular, when $π_2(M)=0$. The Arnold conjecture asserts that the number of fixed points of a Hamiltonian symplectomorphism of $M$ is greater than or equal to the number of critical points of some function on $M$. A modified form of the conjecture, replacing the latter quantity (via Lusternik-Schnirelmann theory) by $cup(M) + 1$, has been proved recentl…

A remark on the Brylinski conjecture for orbifolds

We present reformulation of Mathieu's result on representing cohomology classes of symplectic manifold with symplectically harmonic forms. We apply it to the case of foliated manifolds with transversally symplectic structure and to symplectic orbifolds. We obtain in particular that such representation is always possible for compact Kähler orbifolds.

Symplectic Harmonic theory and the Federer-Fleming deformation theorem

In this article, we initiate a geometric measure theoretic approach to symplectic Hodge theory. In particular, we apply one of the central results in geometric measure theory, the Federer-Fleming deformation theorem, together with the cohomology theory of normal cur- rents on a differential manifold, to establish a fundamental property on symplectic Harmonic forms. We show that on a closed symplectic manifold, every real primitive cohomology class of positive degrees admits a symplectic Harmonic representative not supported on the entire mani- fold. As an application, we use it to investigate …

On the Flux Conjectures

The ``Flux conjecture'' for symplectic manifolds states that the group of Hamiltonian diffeomorphisms is C^1-closed in the group of all symplectic diffeomorphisms. We prove the conjecture for spherically rational manifolds and for those whose minimal Chern number on 2-spheres either vanishes or is large enough. We also confirm a natural version of the Flux conjecture for symplectic torus actions. In some cases we can go further and prove that the group of Hamiltonian diffeomorphisms is C^0-closed in the identity component of the group of all symplectic diffeomorphisms.

The Weinstein Conjecture for Hamiltonian Fibrations

In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurfaces of contact type, under certain technical conditions. The proof is based on the product formula for Gromov-Witten invariants ($GW$-invariant) of Hamiltonian fibrations derived in \cite{H}.

奇异辛流形的 Arnold 猜想

The Arnold conjecture for singular symplectic manifolds

本文通过把奇异辛结构关联到光滑辛形式，证明多类 $b^m$-辛流形的 Arnold 猜想，并给出依赖拓扑的周期 Hamilton 轨道数下界。

抛物系统以及Yang-Mills流和Ricci流的Thom梯度猜想

Thom's gradient conjecture for parabolic systems and the Yang-Mills and Ricci flows

本文将Thom梯度猜想的成立范围从Rn上解析函数的梯度流拓展到希尔伯特空间上含规范对称性的抛物流，验证其对Yang-Mills流、满足条件的Ricci流均成立，还小幅改进了2021年的原始工作。

Taming symplectic forms and the Calabi-Yau equation

We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

Geometric quantization of symplectic maps and Witten's asymptotic conjecture

We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact Kähler manifolds under deformations of the complex structure. We show that the parallel transport in the induced vector bundle over the deformation space behaves like a Toeplitz operator, and compute its first coefficient. We then use this result to establish a semi-classical trace formula for the induced quantization of symplectic maps, and give an application to Witten's asymptotic expansion conjecture for the quantum representations of t…

共形辛Weinstein猜想

A conformal symplectic Weinstein conjecture

本文提出并部分验证了局部共形辛流形上的Weinstein猜想推广，引入Reeb 2-曲线，并利用Gromov-Witten理论构造Fuller指标，获得经典Reeb动力学新结果。