87 research outputs found

### Symplectic fixed points and Lagrangian intersections on weighted projective spaces

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 $q_i\in {\bf q}=\{q_1,..., q_{n+1}\}$ is odd.Comment: Latex, 14 page

### An explicit isomorphism between Floer homology and quantum homology

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

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

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

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
($\pi_1$-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

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 $C^2$-smooth
functionals. It obstructs one using Morse theory to study most of variational
problems of form $F(u)=\int_\Omega f(x, u,..., D^mu)dx$ 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 $C^1$) on a
Hilbert space $H$ which have higher smoothness (but lower than $C^2$) on a
densely and continuously imbedded Banach space $X\subset H$ near a critical
point lying in $X$. (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 $H$ and $X$ 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

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

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 $H^1$-curves

We establish the splitting lemmas (or generalized Morse lemmas) for the
energy functionals of Finsler metrics on the natural Hilbert manifolds of
$H^1$-curves around a critical point or a critical $\R^1$ 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

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 $S^1$-action.Comment: Final versio

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