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Wenmin Gong

Publications and source records attributed to Wenmin Gong.

12 recordsLinked to original sources
arXiv2026

On Cap-Decorated Floer Persistence modules

For a closed symplectically aspherical manifold, we introduce cap-decorated Floer persistence by refining the action-filtered Hamiltonian Floer persistence module with layers defined by iterated actions of homogeneous cohomology ideals. We prove functoriality and stability of these layers, and construct a universal cap-profile pseudodistance bounded above by Hofer distance. As an application, we obtain new invariants of Hamiltonian conjugacy classes, detecting symplectic mapping-class displacement phenomena invisible to ordinary Floer barcodes and spectral invariants. In particular, for a genus-two surface we construct Hamiltonian diffeomorphisms with identical ordinary Floer data but positive separation in the Hamiltonian-conjugacy quotient.

math.SG↗
arXiv2026

Persistent Entropy of Floer Persistence Barcodes

Floer persistence barcodes provide a quantitative way to encode action-filtered Floer homology. Inspired by the Shannon entropy of persistence barcodes in topological data analysis, we introduce a Floer-theoretic entropy invariant, called \textit{persistent entropy}, which measures the asymptotic linear growth rate, under iteration, of the Shannon entropy determined by the distribution of finite bar lengths. This is complementary to the barcode entropy of Çineli--Ginzburg--Gürel, which records the exponential growth rate of the number of not-too-short bars. We prove that, for Hamiltonian diffeomorphisms, the relative and absolute persistent entropies coincide with the corresponding barcode entropies. For Liouville domains, we prove general comparison inequalities and a subexponential length-growth criterion which gives equality beyond the case of vanishing symplectic homology. We also compute the persistent entropy of cotangent disk bundles of negatively curved manifolds and relate it to the topological entropy of the geodesic flow. In addition, we prove Hofer-stability estimates for finite-level Shannon entropy and derive flexibility and rigidity-type questions for barcode and persistent entropies of Reeb flows.

math.SG↗
arXiv2026

Dynamics of composite symplectic Dehn twists

This paper appears as the confluence of hyperbolic dynamics, symplectic topology and low dimensional topology, etc. We show that composite symplectic Dehn twists have certain form of nonuniform hyperbolicity: it has positive topological entropy as well as two families of local stable and unstable Lagrangian manifolds, which are analogous to signatures of pseudo{-}Anosov mapping classes. Moreover, we show that the rank of the Floer cohomology group of these compositions grows exponentially under iterations, and provide a classification of the symplectic mapping class group of the $A^2_m$ configuration, which partially answers a question of Smith concerning the classification of symplectic mapping class group in higher dimensions. Finally, we propose a conjecture on the positive metric entropy of our model and point out its relationship with the standard map.

math.DS↗
arXiv2025

Degenerate symplectic fixed points and Gromov-Witten invariants

We establish a connection between Gromov-Witten invariants and the number of fixed points of Hamiltonian diffeomorphisms on a closed rational symplectic manifold via deformed Hamiltonian spectral invariants. We generalize Givental's symplectic fixed point theorem for Fano toric manifolds to closed rational symplectic manifolds which admit nonzero Gromov-Witten invariants with fixed marked points and one point insertion. We prove a new cuplength estimate of symplectic fixed points involved in deformed spectral invariants. We extend Schwarz's quantum cuplength to the notion of deformed quantum cuplength for symplectic periods and employ it to estimate the number of fixed points of Hamiltonian diffeomorphisms on monotone symplectic manifolds with nonzero mixed Gromov-Witten invariants.

math.SG↗
arXiv2024

Lagrangian intersections and a conjecture of Arnol'd

We prove a degenerate homological Arnol'd conjecture on Lagrangian intersections beyond the case studied by A. Floer and H. Hofer via a new version of Lagrangian Ljusternik--Schnirelman theory. We introduce the notion of (Lagrangian) fundamental quantum factorizations and use them to give some uniform lower bounds of the numbers of Lagrangian intersections for some classical examples including Clifford tori in complex projective spaces. Additionally, we use the Lagrangian Ljusternik-Schnirelman theory to study the size of the intersection of a monotone Lagrangian with its image of a Hamiltonian diffeomorphism.

math.SG↗
arXiv2024

The unbounded Lagrangian spectral norm and wrapped Floer cohomology

We investigate the question of whether the spectral metric on the orbit space of a fiber in the disk cotangent bundle of a closed manifold, under the action of the compactly supported Hamiltonian diffeomorphism group, is bounded. We utilize wrapped Floer cohomology to define the spectral invariant of an admissible Lagrangian submanifold within a Weinstein domain. We show that the pseudo-metric derived from this spectral invariant is a valid $Ham$-invariant metric. Furthermore, we establish that the spectral metric on the orbit space of an admissible Lagrangian is bounded if and only if the wrapped Floer cohomology vanishes. Consequently, we prove that the Lagrangian Hofer diameter of the orbit space for any fiber in the disk cotangent bundle of a closed manifold is infinite.

math.SG↗
arXiv2022

Minimax periodic orbits of convex Lagrangian systems on complete Riemannian manifolds

In this paper we study the existence of periodic orbits with prescribed energy levels of convex Lagrangian systems on complete Riemannian manifolds. We extend the existence results of Contreras by developing a modified minimax principal to a class of Lagrangian systems on noncompact Riemannian manifolds, namely the so called $\lsh$ Lagrangian systems. In particular, we prove that for almost every $k\in(0,c_u(L))$ the exact magnetic flow associated to a $\lsh$ Lagrangian has a contractible periodic orbit with energy $k$. We also discuss the existence and non-existence of closed geodesics on the product Riemannian manifold $\R\times M$.

math.DG↗
arXiv2020

Floer homology in the cotangent bundle of a closed Finsler manifold and noncontractible periodic orbits

We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. We generalize the BPS capacities and earlier constructions of Weber (2006 Duke Math. J. 133, 527-568) and other authors Biran et al (2003 Duke Math. J. 119, 65-118) to the Finsler setting. We then obtain a number of applications including: (1) generalizing the main theorem of Xue (2017 J. Symplectic Geom. 15, 905-936) to the Lie group setting, (2) preservation of minimal Finsler length of closed geodesics in any given free homotopy class by symplectomorphisms, (3) existence of periodic orbits for Hamiltonian systems separating two Lagrangian submanifolds, (4) existence of periodic orbits for Hamiltonians on noncompact domains, (5) existence of periodic orbits for Lorentzian Hamiltonian in higher dimensional case, (6) partial solution to a conjecture of Kawasaki (2016 Heavy subsets and non-contractible trajectories (arXiv:1606.01964)), (7) results on squeezing/nonsqueezing theorem on torus cotangent bundles, etc.

math.SG↗
arXiv2019

Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

In this paper we establish the existence of periodic orbits belonging to any $σ$-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact $2$-form $σ$ admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any nontrivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran-Polterovich-Salamon capacity.

math.SG↗
arXiv2016

Existence results for coupled Dirac systems via Rabinowitz-Floer theory

In this paper, we construct the Rabinowitz-Floer homology for the coupled Dirac system \begin{equation*} \left\{ \begin{aligned} Du=\frac{\partial H}{\partial v}(x,u,v)\hspace{4mm} {\rm on} \hspace{2mm}M,\\ Dv=\frac{\partial H}{\partial u}(x,u,v)\hspace{4mm} {\rm on} \hspace{2mm}M, \end{aligned} \right. \end{equation*} where $M$ is an $n$-dimensional compact Riemannian spin manifold, $D$ is the Dirac operator on $M$, and $H:ΣM\oplus ΣM\to \mathbb{R}$ is a real valued superquadratic function of class $C^1$ with subcritical growth rates. Solutions of this system can be obtained from the critical points of a Rabinowitz-Floer functional on a product space of suitable fractional Sobolev spaces. In particular, we consider the $S^1$-equivariant $H$ that includes a nonlinearity of the form $$ H(x,u,v)=f(x)\frac{|u|^{p+1}}{p+1}+g(x)\frac{|v|^{q+1}}{q+1}, $$ where $f(x)$ and $g(x)$ are strictly positive continuous functions on $M$, and $p>1,q>1$ satisfy $$ \frac{1}{p+1}+\frac{1}{q+1}>\frac{n-1}{n}. $$ We establish the existence of a nontrivial solution by computing the Rabinowitz-Floer homology in the Morse-Bott situation.

math.AP↗