87 research outputs found

    Symplectic fixed points and Lagrangian intersections on weighted projective spaces

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    In this note we observe that Arnold conjecture for the Hamiltonian maps still holds on weighted projective spaces \CP^n({\bf q}), and that Arnold conjecture for the Lagrange intersections for (\CP^n({\bf q}), \RP^n({\bf q})) is also true if each weight qi∈q={q1,...,qn+1}q_i\in {\bf q}=\{q_1,..., q_{n+1}\} is odd.Comment: Latex, 14 page

    An explicit isomorphism between Floer homology and quantum homology

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    We use Liu-Tian's virtual moduli cycle methods to construct detailedly the explicit isomorphism between Floer homology and quantum homology for any closed symplectic manifold that was first outlined by Piunikhin, Salamon and Schwarz for the case of the semi-positive symplectic manifolds.Comment: Latex, published versio

    Some critical point theorems and applications

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    This paper is a continuation of \cite{Lu1}. In Part I, applying the new splitting theorems developed therein we generalize previous some results on computations of critical groups and some critical point theorems to weaker versions. In Part II (in progress), they are used to study multiple solutions for nonlinear higher order elliptic equations described in the introduction of \cite{Lu1}.Comment: Latex, 31 page

    The Weinstein conjecture in the uniruled manifolds

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    In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.Comment: 5 pages, LaTe

    Finiteness of the Hofer-Zehnder capacity of neighborhoods of symplectic submanifolds

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    We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite (Ο€1\pi_1-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds near closed symplectic submanifolds in any symplectic manifold.Comment: 33 page

    The splitting lemmas for nonsmooth functionals on Hilbert spaces I

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    The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least C2C^2-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form F(u)=∫Ωf(x,u,...,Dmu)dxF(u)=\int_\Omega f(x, u,..., D^mu)dx as in (\ref{e:1.1}). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than C1C^1) on a Hilbert space HH which have higher smoothness (but lower than C2C^2) on a densely and continuously imbedded Banach space XβŠ‚HX\subset H near a critical point lying in XX. (This splitting theorem generalize almost all previous ones to my knowledge). Moreover, a new theorem of Poincar\'e-Hopf type and a relation between critical groups of the functional on HH and XX are given. Different from the usual implicit function theorem method and dynamical system one our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai \cite{DHK} with some techniques from \cite{JM, Skr, Va1}. Our theory is applicable to the Lagrangian systems on compact manifolds and boundary value problems for a large class of nonlinear higher order elliptic equations.Comment: 68 pages; v2: for the published version we correct a few typo and state Theorems A.1, A.2 in a more precise wa

    Morse theory methods for a class of quasi-linear elliptic systems of higher order

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    We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type result. They are applicable to a wide range of multiple integrals with quasi-linear elliptic Euler equations and systems of higher order.Comment: To appear in Calculus of Variations and PDE's, arXiv admin note: substantial text overlap with arXiv:1702.0666

    Parameterized splitting theorems and bifurcations for potential operators

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    We show that parameterized versions of splitting theorems in Morse theory can be effectively used to generalize some famous bifurcation theorems for potential operators. In particular, such generalizations based on the author's recent splitting theorems [38, 39, 42, 43] and that of [8] are given though potential operators in [42, 43] have weaker differentiability, even discontinuous. As applications, we obtain many bifurcation results for quasi-linear elliptic Euler equations and systems of higher order.Comment: 73 pages. arXiv admin note: a complete rewriting and large extension for last versio

    Splitting lemmas for the Finsler energy functional on the space of H1H^1-curves

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    We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of H1H^1-curves around a critical point or a critical R1\R^1 orbit of a Finsler isometry invariant closed geodesic. They are the desired generalization on Finsler manifolds of the corresponding Gromoll-Meyer's splitting lemmas on Riemannian manifolds (\cite{GM1, GM2}). As an application we extend to Finsler manifolds a result by Grove and Tanaka \cite{GroTa78, Tan82} about the existence of infinitely many, geometrically distinct, isometry invariant closed geodesics on a closed Riemannian manifold.Comment: 56 pages, Latex, latest version, to appear in Proceedings of the London Mathematical Societ

    Symplectic capacities of toric manifolds and related results

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    In this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds with S1S^1-action.Comment: Final versio
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