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An-Min Li

Publications and source records attributed to An-Min Li.

At least 19 recordsLinked to original sources
arXiv2020

On the mean curvature flow of submanifolds in the standard Gaussian space $^†$

In this paper, we study the regular geometric behavior of the mean curvature flow (MCF) of submanifolds in the standard Gaussian metric space $({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$ where $({\mathbb R}^{m+p},\ol g)$ is the standard Euclidean space and $x\in{\mathbb R}^{m+p}$ denotes the position vector. Note that, as a special Riemannian manifold, $({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$ has an unbounded curvature. Up to a family of diffeomorphisms on $M^m$, the mean curvature flow we considered here turns out to be equivalent to a special variation of the ``{\em conformal mean curvature flow}\,'' which we have introduced previously. The main theorem of this paper indicates, geometrically, that any immersed compact submanifold in the standard Gaussian space, with the square norm of the position vector being not equal to $m$, will blow up at a finite time under the mean curvature flow, in the sense that either the position or the curvature blows up to infinity; Moreover, by this main theorem, the interval $[0,T)$ of time in which the flowing submanifolds keep regular has some certain optimal upper bound, and it can reach the bound if and only if the initial submanifold either shrinks to the origin or expands uniformly to infinity under the flow. Besides the main theorem, we also obtain some other interesting conclusions which not only play their key roles in proving the main theorem but also characterize in part the geometric behavior of the flow, being of independent significance.

math.DG↗
arXiv2018

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

Let $f$ be a smooth plurisubharmonic function which solves $$ \det(f_{i\bar j})=1\;\;\;\;\;\;\mbox{in }Ω\subset \mathbb C^n.$$ Suppose that the metric $ω_{f}=\sqrt{-1}f_{i\bar j}dz_{i}\wedge d\bar z_{j}$ 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.

math.DG↗
arXiv2018

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.

math.SG↗
arXiv2016

Prescribed Scaler Curvatures for Homogeneous Toric Bundles

In this paper, we study the generalized Abreu equation on a Delzant ploytope $Δ\subset \mathbb{R}^2$ and prove the existence of the constant scalar metrics of homogeneous toric bundles under the assumption of an appropriate stability.

math.DG↗
arXiv2015

Extremal metrics on toric surfaces

In this paper, we study the Abreu equation on toric surfaces. In particular, we prove the existence of the positive extremal metric when relative $K$-stability is assumed.

math.DG↗
arXiv2015

Gluing principle for orbifold stratified spaces

In this paper, we explore the theme of orbifold stratified spaces and establish a general criterion for them to be smooth orbifolds. This criterion utilizes the notion of linear stratification on the gluing bundles for the orbifold stratified spaces. We introduce a concept of good gluing structure to ensure a smooth structure on the stratified space. As an application, we provide an orbifold structure on the coarse moduli space $\bar{M}_{g, n}$ of stable genus $g$ curves with $n$-marked points. Using the gluing theory for $\bar{M}_{g, n} $ associated to horocycle structures, there is a natural orbifold gluing atlas on $\bar{M}_{g, n} $. We show this gluing atlas can be refined to provide a good orbifold gluing structure and hence a smooth orbifold structure on $\bar{M}_{g,n}$. This general gluing principle will be very useful in the study of the gluing theory for the compactified moduli spaces of stable pseudo-holomorphic curves in a symplectic manifold.

math.GT↗
arXiv2015

Relative Invariants, Contact Geometry and Open String Invariants

In this paper we propose a theory of contact invariants and open string invariants, which are generalizations of the relative invariants. We introduce two moduli spaces $\bar{\mathcal{M}}_{A}(M^{+},C,g,m+ν,{\bf y},{\bf p},(\mathbf{k},\mathfrak{e}))$ and $\bar{\mathcal{M}}_{A}(M,L;g,m+ν,{\bf y},{\bf p},\overrightarrowμ)$, prove the compactness of the moduli spaces and the existence of the invariants.

math.SG↗
arXiv2014

Virtual neighborhood technique for pseudo-holomorphic spheres

This is the first part of a trilogy where we apply the theory of virtual manifold/orbifolds developed by the first named author and Tian to study the Gromov-Witten moduli spaces. In this paper, we resolve the main analytic issue arising from the lack of differentiability of $PSL(2, \C)$-action on spaces of $W^{1, p}$-maps from the Riemann sphere to a symplectic manifold $(X, ω, J)$ with a non-zero homology class $A$. In particular, we establish the slice and tubular neighbourhood theorems for $PSL(2, \C)$-action along smooth maps, and construct a $PSL(2, \C)$-obstruction bundle along $PSL(2, \C)$-orbit of a pseudo-holomorphic map representing a point in the moduli space $\cM_{0, 0}(X, A)$. In Sections 2 and 3 of this paper, we explain an integration theory on virtual orbifolds using proper étale groupoids and establish the virtual neighborhood technique for a general orbifold Fredholm system. When the moduli space $\cM_{0, 0}(X, A)$ of pseudo-holomorphic spheres in $(X, ω, J)$ is compact, applying the virtual neighborhood technique developed in Section 3, we obtain a virtual system for the moduli space $\cM_{0, 0}(X, A)$ of pseudo-holomorphic spheres in $(X, ω, J)$ and show that the genus zero Gromov-Witten invariant is well-defined.

math.GT↗