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An-Min Li

Publications and source records attributed to An-Min Li.

35 records · Page 2Linked to original sources
arXiv2014

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].

math.SG↗
arXiv2012

E-Characteristic Polynomials of Tensors

In this paper, we show that the coefficients of the E-characteristic polynomial of a tensor are orthonormal invariants of that tensor. When the dimension is 2, some simplified formulas of the E-characteristic polynomial are presented. A re- sultant formula for the constant term of the E-characteristic polynomial is given. We then study the set of tensors with infinitely many eigenpairs and the set of irregular tensors, and prove both the sets have codimension 2 as subvarieties in the projective space of tensors. This makes our perturbation method workable. By using the perturbation method and exploring the difference between E-eigenvalues and eigenpair equivalence classes, we present a simple formula for the coefficient of the leading term of the E-characteristic polynomial, when the dimension is 2.

math.SP↗
arXiv2011

Notes on Chern's Affine Bernstein Conjecture

There were two famous conjectures on complete affine maximal surfaces, one due to E. Calabi, the other to S.S. Chern. Both were solved with different methods about one decade ago by studying the associated Euler-Lagrange equation. Here we survey two proofs of Chern's conjecture in our recent monograph [L-X-S-J], in particular we add some details of the proofs of auxiliary material that were omitted in [L-X-S-J]. We describe the related background in our Introduction. Our survey is suitable as a report about recent developments and techniques in the study of certain Monge-Ampere equations.

math.DG↗
arXiv2010

Complete Affine K$\ddot{a}$hler Manifolds

In this paper we prove that for a complete, connected and oriented Käler affine manifold $(M,G)$ of dimension $n,$ if it is Kähler affine Ricci flat or the K$\ddot{a}$hler affine scalar curvature $S\equiv0,$ ($n\leq 5$), then the universal covering manifold $\widetilde{M}$ of $M$ is isometric to the Euclidean n-space $E^{n}$.

math.DG↗
arXiv2008

Singular symplectic flops and Ruan cohomology

In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient $$ W_r=\{(x,y,z,t)|xy-z^{2r}+t^2=0 \}/μ_r(a,-a,1,0), r\geq 1, $$ which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let $X$ and $Y$ be two symplectic orbifolds connected by such a flop. We study orbifold Gromov-Witten invariants of exceptional classes on $X$ and $Y$ and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan.

math.SG↗
arXiv2007

A Rigidity Theorem for Affine Kähler-Ricci Flat Graph

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

math.DG↗
arXiv2000

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.

math.AG↗
arXiv1999

The number of ramified covering of a Riemann surface by Riemann surface

Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann surface by Riemann surface with elementary branch points and prescribed ramification type over a special point.

math.AG↗