記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：arXiv  

We study Farber's topological complexity for monotone symplectic manifolds.
More precisely, we estimate the topological complexity of 4-dimensional
spherically monotone manifolds whose Kodaira dimension is not $-\infty$.

DOI： 10.48550/ARXIV.2311.14989

arXiv

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：Mathematical Society of Japan (Project Euclid)  

DOI： 10.2969/jmsj/84278427

arXiv

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：World Scientific Pub Co Pte Lt  

In this paper, we introduce the notion of pseudoheaviness of closed subsets of closed symplectic manifolds and prove the existence of pseudoheavy fibers of moment maps. In particular, we generalize Entov and Polterovich’s theorem, which ensures the existence of non-displaceable fibers. As its application, we provide a partial answer to a problem posed by them, which asks the existence of heavy fibers. Moreover, we obtain a family of singular Lagrangian submanifolds in [Formula: see text] with various rigidities.

DOI： 10.1142/s0219199720500479

arXiv

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記述言語：英語   掲載種別：研究論文（学術雑誌）  

We show that the presence of a non-contractible one-periodic orbit of a
Hamiltonian diffeomorphism of a connected closed symplectic manifold
$(M,\omega)$ implies the existence of infinitely many non-contractible simple
periodic orbits, provided that the symplectic form $\omega$ is aspherical and
the fundamental group $\pi_1(M)$ is either a virtually abelian group or an
$\mathrm{R}$-group. We also show that a similar statement holds for Hamiltonian
diffeomorphisms of closed monotone or negative monotone symplectic manifolds
under the same conditions on their fundamental groups. These results generalize
some works by Ginzburg and G\"urel. The proof uses the filtered Floer--Novikov
homology for non-contractible periodic orbits.

DOI： 10.4310/JSG.2019.v17.n6.a9

arXiv

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：World Scientific Pub Co Pte Lt  

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding [Formula: see text] with respect to the fragmentation norm on the group [Formula: see text] of Hamiltonian diffeomorphisms of a symplectic manifold [Formula: see text]. As an application, we provide an answer to Brandenbursky’s question on fragmentation norms on [Formula: see text], where [Formula: see text] is a closed Riemannian surface of genus [Formula: see text].

DOI： 10.1142/S179352532050017X

arXiv

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記述言語：日本語

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Many transformation groups on manifolds are simple, but their universal
coverings are not. In the present paper, we study the concept of relatively
simple group, that is, a group with the maximum proper normal subgroup. We show
that many examples of universal coverings of transformation groups are
relatively simple, including the universal covering
$\widetilde{\mathrm{Ham } }(M,\omega)$ of the group of Hamiltonian
diffeomorphisms of a closed symplectic manifold $(M,\omega)$.
Tsuboi constructed a metric space $\mathcal{M}(G)$ for a simple group $G$. We
generalize his construction to relatively simple groups, and study their large
scale geometric structure. In particular, Tsuboi's metric space of
$\widetilde{\mathrm{Ham } }(M, \omega)$ is not quasi-isometric to the half line
for every closed symplectic manifold $(M,\omega)$.

arXiv

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その他リンク： http://arxiv.org/pdf/2412.00839v1

記述言語：英語   出版者・発行元：arXiv  

In this paper, we show that the pointwise finite-dimensional two-parameter
persistence module $\mathbb{HF}_*^{(\bullet,\bullet]}$, defined in terms of
interlevel filtered Floer homology, is rectangle-decomposable. This allows for
the definition of a barcode $\mathcal{B}_*^{(\bullet,\bullet]}$ consisting only
of rectangles in $\mathbb{R}^2$ associated with
$\mathbb{HF}_*^{(\bullet,\bullet]}$. We observe that this rectangle barcode
contains information about Usher's boundary depth and spectral invariants
developed by Oh, Schwarz, and Viterbo. Moreover, we introduce a novel invariant
extracted from $\mathcal{B}_*^{(\bullet,\bullet]}$, which proves instrumental
in detecting periodic solutions of Hamiltonian systems. Additionally, we
establish relevant stability results, particularly concerning the bottleneck
distance and Hofer's distance.

DOI： 10.48550/ARXIV.2312.07847

arXiv

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その他リンク： http://arxiv.org/pdf/2312.07847

対象： 高校生

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