Perturbed closed geodesics are periodic orbits: Index and Transversality

扰动特殊拉格朗日子流形的横截性

Transversality for perturbed special Lagrangian submanifolds

本文在配备广义Calabi-Yau结构的6维流形上，证明了扰动特殊拉格朗日子流形模空间的横截性定理，并表明该模空间一般由孤立点组成，为Donaldson-Segal计数方案提供了关键理论支撑。

Perturbed geodesics on the moduli space of flat connections and Yang-Mills theory

If we consider the moduli space of flat connections of a non trivial principal SO(3)-bundle over a surface, then we can define a map from the set of perturbed closed geodesics, below a given energy level, into families of perturbed Yang-Mills connections depending on a small parameter. In this paper we show that this map is a bijection and maps perturbed geodesics into perturbed Yang-Mills connections with the same Morse index.

Morse theory for the Hofer length functional

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: Ω\text {Ham}(M, ω) \to \mathbb{R}$, with Gromov-Witten/Floer theory, for monotone symplectic manifolds $ (M, ω) $. This gives some immediate restrictions on the topology of the group of Hamiltonian symplectomorphisms (possibly relative to the Hofer length functional), and a criterion for non-existence of certain higher index geodesics for the Hofer length functional. The argument is based on a certain automatic transversality p…

闭Finsler与Riemann曲面上的等分布闭测地线

Equidistributed Closed Geodesics on Closed Finsler and Riemannian Surfaces

本文利用嵌入切触同调的体积性质、局部变分构造与横截性论证，证明了闭曲面上一般Finsler、对称Finsler及Riemann度量存在非退化闭测地线的等分布序列。

紧致模空间的抽象扰动定理

An Abstract Perturbation Theorem for Compact Moduli Spaces

针对Fredholm截面的紧致零点集，提出一种抽象扰动方法，在保持已有横截性区域不变的前提下构造邻近的紧致光滑流形，并应用于证明Hamilton作用泛函在辛非球面流形上不同作用值临界点的存在性。

Heat flow on the moduli space of flat connections and Yang-Mills theory

It is known that there is a bijection between the perturbed closed geodesics, below a given energy level, on the moduli space of flat connections M and families of perturbed Yang-Mills connections depending on a small parameter. In this paper we study the heat flow on the loop space on M and the Yang-Mills L^2-flows for a 3-manifold N with partial rescaled metrics. Our main result is that the bounded Morse homology of the loop space on M is isomorphic to the bounded Morse homologies of the connections space of N.