Remark on symplectic relative Gromov-Witten invariants and degeneration formula

In this note, we give item-by-item responses to the criticisms raised in [TZ] by Tehrani ad Zinger on our paper [LR]. We illuminate the main ideas and contributions in [LR] in section 2, itemize the responses to issues raised in [TZ] and conclude that we have provided a complete proof of the degeneration formula in our published paper [LR] and its more detailed versions in arXiv. In [TZ], the authors made an effort in comparing the methods and ideas in [LR] vs [IP-1] [IP-2], but their criticisms on [LR] are based on their own lack of sufficient understanding of [LR].Comment: 20 page

A Liouville Theorem on the PDE det⁡(fijˉ)=1\det(f_{i\bar j})=1

Let $f$ be a smooth plurisubharmonic function which solves \det(f_{i\bar j})=1\;\;\;\;\;\;\mbox{in }\Omega\subset \mathbb C^n. Suppose that the metric $\omega_{f}=\sqrt{-1}f_{i\bar j}dz_{i}\wedge d\bar z_{j}$ is complete and $f$ satisfies the growth condition $$C^{-1}(1+|z|^2)\leq f\leq C(1+ |z|^2),\;\;\;\; as\;\;\; |z|\to \infty.$$ for some $C>0,$ then $f$ is quadratic

The Exponential Decay of Gluing Maps for JJ-Holomorphic map Moduli Spaces

We prove the exponential decay of the derivative of the gluing maps with respect to the gluing parameter.Comment: v3 title changed, 36pages. v2 26pages, minor revision. v1 25pages. Welcome comment

A Finite Rank Bundle over JJ-Holomorphic map Moduli Spaces

We study a finite rank bundle $\mathbf{F}$ over a neighborhood of $J$-Holomorphic map Moduli Spaces, prove the exponential decay of the derivative of the gluing maps for $\mathbf{F}$ with respect to the gluing parameter.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1710.10581, arXiv:1506.0633

Contact Invariants, Open String Invariants and Weinstein Conjecture

We propose a theory of contact invariants and open string invariants, assuming that the almost complex $J$ is either non-degenerate or of Bott-type. We do not choose the complex structure $\tilde{J}$ such that $L_X\tilde{J}=0$ on periodic orbits.Comment: 13pages. arXiv admin note: substantial text overlap with arXiv:1501.0109

Virtual Neighborhood Technique for Holomorphic Curve Moduli Spaces

In this paper we use the approach of Ruan and Li-Ruan to construct virtual neighborhoods and show that the Gromov-Witten invariants can be defined as an integral over top strata of virtual neighborhood. We prove that the invariants defined in this way satisfy all the Gromov-Witten axioms of Kontsevich and Manin.Comment: Any comments are welcome. 44 pages,(47pages,v2

Quantum mechanics in the general quantum systems (V): Hamiltonian eigenvalues

We derive out a complete series expression of Hamiltonian eigenvalues without any approximation and cut in the general quantum systems based on Wang's formal framework \cite{wang1}. In particular, we then propose a calculating approach of eigenvalues of arbitrary Hamiltonian via solving an algebra equation satisfied by a kernal function, which involves the contributions from all order perturbations. In order to verify the validity of our expressions and reveal the power of our approach, we calculate the ground state energy of a quartic anharmonic oscillator and have obtained good enough results comparing with the known one.Comment: 18 pages, No figure. This is the fifth manuscript. Previous manuscripts see arXiv:quant-ph/0611216, arXiv:quant-ph/0611217, arXiv:quant-ph/0601051 and arXiv:quant-ph/061206

Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds I

We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete formula for the change of GW-invariants of any genus transform under flop and a general type I extremal transition. Other extremal transition will be handled in a subsequent paper.Comment: Latex, revised version, a much shorter and better written versio

Symplectic virtual localization of Gromov-Witten invariants

We show that moduli spaces of stable maps admits virtual orbifold structure. The symplectic version of virtual localization formula is obtained.Comment: 62 page

A Rigidity Theorem for Affine K\"ahler-Ricci Flat Graph

It is shown that any smooth strictly convex global solution of $$\det(\frac{\partial^{2}u}{\partial \xi_{i}\partial \xi_{j}}) = \exp \left\{-\sum_{i=1}^n d_i \frac{\partial u}{\partial \xi_{i}} - d_0\right\},$$ where $d_0$, $d_1$,...,$d_n$ are constants, must be a quadratic polynomial. This extends a well-known theorem of J\"{o}rgens-Calabi-Pogorelov.Comment: 24 page