arXiv subjects

Guangcun Lu

Publications and source records attributed to Guangcun Lu.

46 records · Page 3Linked to original sources

arXiv2006

Gromov-Witten invariants and pseudo symplectic capacities

We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions, then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic uniruled manifolds (including the uniruled manifolds in algebraic geometry) and also show that any closed symplectic submanifold of codimension two in any symplectic manifold has a small neighborhood whose Hofer-Zehnder capacity is less than a given positive number. Finally, we give two results on symplectic packings in Grassmannians and on Seshadri constants.

math.SG↗

arXiv2006

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 ($π_1$-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds near closed symplectic submanifolds in any symplectic manifold.

math.SG↗

arXiv2006

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.

math.SG↗

arXiv2006

Virtual moduli cycles and Gromov-Witten invariants of noncompact symplectic manifolds

This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get solutions of the generalized string equation and dilation equation and their variants. The more solutions of WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To realize these purposes we further develop the language introduced by Liu-Tian to describe the virtual moduli cycle (defined by Liu-Tian, Fukaya-Ono, Li-Tian, Ruan and Siebert).

math.DG↗

arXiv2006

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.

math.SG↗

arXiv2004

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.

math.DG↗

arXiv2004

An extension of Biran's Lagrangian barrier theorem

We use the Gromov-Witten invariants and a nonsqueezing theorem by the author to affirm a conjecture by P.Biran on the Lagrangian barriers.

math.SG↗

arXiv2003

Gromov-Witten invariants and rigidity of Hamiltonian loops with compact support on noncompact symplectic manifolds

In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian's method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the transversality problems in the case of noncompact manifolds. Moreover, the techniques are also used to study the topological rigidity of Hamiltonian loops with compact support on a class of noncompact symplectic manifolds.

math.DG↗

arXiv2000

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.

math.DG↗

arXiv2000

Periodic motion of a charge on a manifold in the magnetic fields

In this paper we prove the existence of a periodic motion of a charge on a large class of manifolds under the action of the magnetic fields. Our methods also give a class of closed manifolds whose cotangent bundle contain no the closed exact Lagrangian submanifolds.

math.DG↗