Anna Gori, Fabio Podesta'
We deal with compact Kähler manifolds $M$ acted on by a compact Lie group $K$ of isometries, whose complexification $K^\C$ has exactly one open and one closed orbit in $M$. If the $K$-action is Hamiltonian, we obtain results on the cohomology and the $K$-equivariant cohomology of $M$.
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We compare Lagrangian thimbles for the potential of a Landau-Ginzburg model to the Morse theory of its real part. We explore Landau-Ginzburg models defined using Lie theory, constructing their real Lagrangian thimbles explicitly and comparing them to the stable and unstable manifolds of the real gradient flow.
This is an expository paper devoted to the Morse index of Chaperon's generating families of Hamiltonian diffeomorphisms. After reviewing the construction of such generating families, we present Bott's iteration theory in this setting: we study how the Morse index of a critical point corresponding to an iterated periodic orbit depends on the order of iteration of the orbit. We also investigate the precise dependence of the Morse index from the choice of the generating family associated to a given Hamiltonian diffeomorphism, which will allow to see the Morse index as a Maslov index for the linea…
We classify generic coadjoint orbits for symplectomorphism groups of compact symplectic surfaces with or without boundary. We also classify simple Morse functions on such surfaces up to a symplectomorphism.
Floer theory was originally devised to estimate the number of 1-periodic orbits of Hamiltonian systems. In earlier works, we constructed Floer homology for homoclinic orbits on two dimensional manifolds using combinatorial techniques. In the present paper, we study theoretic aspects of computational complexity of homoclinic Floer homology. More precisely, for finding the homoclinic points and immersions that generate the homology and its boundary operator, we establish sharp upper bounds in terms of iterations of the underlying symplectomorphism. This prepares the ground for future numerical w…
A Morse-Bott unification of the Grassmannians of a symplectic vector space
本文提出一种基于相容复结构的二次莫尔斯-博特函数，统一了辛向量空间的各类格拉斯曼流形，并通过负梯度流将其退化为单位群的紧致齐性空间。
We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between linearized contact homology of a fillable contact manifold and symplectic homology of its filling.
Given a contact three manifold $Y$ with a nondegenerate contact form $λ$, and an almost complex structure $J$ compatible with $λ$, its embedded contact homology $ECH(Y,λ)$ is defined (arXiv:1303.5789) and only depends on the contact structure. In this paper we explain how to compute ECH for Morse-Bott contact forms whose Reeb orbits appear in $S^1$ families, assuming the almost complex structure $J$ can be chosen to satisfy certain transversality conditions (this is the case for instance for boundaries of concave or convex toric domains, or if all the curves of ECH index one have genus zero). …
A point q in a contact manifold is called a translated point for a contactomorphism ϕ, with respect to some fixed contact form, if ϕ(q) and q belong to the same Reeb orbit and the contact form is preserved at q. In this article we discuss a version of the Arnold conjecture for translated points of contactomorphisms and, using generating functions techniques, we prove it in the case of spheres (under a genericity assumption) and projective spaces.
In this note we study two index questions. In the first we establish the relationship between the Morse indices of the free time action functional and the fixed time action functional. The second is related to Rabinowitz Floer homology. Our index computations are based on a correction term which is defined as follows: around a non-degenerate Hamiltonian orbit lying in a fixed energy level a well-known theorem says that one can find a whole cylinder of orbits parametrized by the energy. The correction term is determined by whether the periods of the orbits are increasing or decreasing as one mo…
Non-minimality and instability of brake orbits for natural Lagrangians on Riemannian manifolds
研究了自然拉格朗日系统在黎曼流形上制动轨道的非最小性和不稳定性，通过莫塞指数分析揭示了其稳定性条件及退化特性。
In this paper we study topological lower bounds on the number of zeros of closed 1-forms without Morse type assumptions. We prove that one may always find a representing closed 1-form having at most one zero. We introduce and study a generalization $cat(X,ξ)$ of the notion of Lusternik - Schnirelman category, depending on a topological space $X$ and a cohomology class $ξ\in H^1(X;\R)$. We prove that any closed 1-form has at least $cat(X,ξ)$ zeros assuming that it admits a gradient-like vector field with no homoclinic cycles. We show that the number $cat(X,ξ)$ can be estimated from below in ter…
The Hamiltonian flow of the standard metric Hamiltonian with respect to the twisted symplectic structure on the cotangent bundle describes the motion of a charged particle on the base. We prove that under certain natural hypotheses the number of periodic orbits on low energy levels for this flow is at least the sum of Betti numbers of the base. The problem is closely related to the existence question for periodic orbits on energy levels of a proper Hamiltonian near a Morse-Bott non-degenerate minimum. In this case, when some extra requirements are met, we also give a lower bound for the number…
We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, which is a generalization of equivariant formality for torus actions without fixed points. As a consequence, a generic component of the contact moment map is a perfect Morse-Bott function for the basic cohomology of the orbit foliation F of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of F is tr…
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