定量托马斯-丘唯一性

Finite Time Singularities for Lagrangian Mean Curvature Flow

Given any embedded Lagrangian on a four dimensional compact Calabi-Yau, we find another Lagrangian in the same Hamiltonian isotopy class which develops a finite time singularity under mean curvature flow. This contradicts a weaker version of the Thomas-Yau conjecture regarding long time existence and convergence of Lagrangian mean curvature flow.

Topology of certain symplectic conifold transitions of $CP^{1}$-bundles

In this paper, we prove the existence of certain symplectic conifold transitions on all $CP^{1}$-bundles over symplectic 4--manifolds, which generalizes Smith, Thomas and Yau's examples of symplectic conifold transitions on trivial $CP^{1}$-bundles over Kähler surfaces. Our main result is to determine the diffeomorphism types of such symplectic conifold transitions of $CP^{1}$-bundles. In particular, this implies that in the case of trivial $CP^{1}$-bundles over projective surfaces, Smith, Thomas and Yau's examples of symplectic conifold transitions are diffeomorphic to Kähler 3--folds.

Hamiltonian mean curvature flow

Let (Σ, ω) be a compact Riemann surface with constant curvature c. In this work, we proved that the mean curvature flow of a given Hamiltonian diffeomorphism on Σ provides a smooth path in Ham(Σ), the group of all Hamiltonian diffeomorphisms of Σ. This result gives a proof, in the case of graph of Hamiltonian diffeomorphisms to the conjecture of Thomas and Yau asserting that the mean curvature flow of a compact embedded Lagrangian submanifold S with zero Maslov class in a Calabi- Yau manifolds M exists for all time and converges smoothly to a special Lagrangian submanifold in the Hamiltonian i…

Lagrangian mean curvature flow in the complex projective plane

We prove a Thomas--Yau-type conjecture for monotone Lagrangian tori satisfying a symmetry condition in the complex projective plane $\mathbb{CP}^2$. We show that such tori exist for all time under Lagrangian mean curvature flow with surgery, undergoing at most a finite number of surgeries before flowing to a minimal Clifford torus in infinite time. Furthermore, we show that we can construct a torus with any finite number of surgeries before convergence. Along the way, we prove many interesting subsidiary results and develop methods which should be useful in studying Lagrangian mean curvature f…

Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow

Let $M$ be a Calabi-Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau math.DG/0104196, math.DG/0104197 conjectured that there should be a notion of "stability" for such $L$, and that if $L$ is stable then Lagrangian mean curvature flow $\{L^t:t\in[0,\infty)\}$ with $L^0=L$ should exist for all time, and $L^\infty=\lim_{t\to\infty}L^t$ should be the unique special Lagrangian in the Hamiltonian isotopy class of $L$. This paper is an attempt to update the Thomas-Yau conjectures, and discuss related issues. It is a folklore conjecture that there exists a Bridgeland …

Supercritical deformed Hermitian-Yang-Mills equation

In this paper, we provide a necessary and sufficient condition for the solvability of the supercritical deformed Hermitian-Yang-Mills equation using integrals on subvarieties. This result confirms the mirror version of the Thomas-Yau conjecture about special Lagrangian submanifolds on Calabi-Yau manifolds.

全纯 Floer 理论与 Donaldson-Thomas 不变量

Holomorphic Floer theory and Donaldson-Thomas invariants

本文提出全纯 Floer 理论的若干预期性质，并猜想其可将 Hitchin/Seiberg-Witten 可积系统与 Donaldson-Thomas 不变量及四维 N=2 理论的 BPS 谱联系起来。

log Calabi-Yau 3-folds 的 Gromov-Witten invariants 是 holomorphic lagrangian correspondences

Gromov-Witten invariants of log Calabi-Yau 3-folds are holomorphic lagrangian correspondences

本文引入了log schemes上的全纯辛范畴，证明log Calabi-Yau 3-folds的Gromov-Witten不变量可自然编码为全纯拉格朗日对应，并据此猜想其与Donaldson-Thomas不变量通过酉拉格朗日对应建立联系。