arXiv subjects
Publications and source records attributed to Xiaoli Han.
In this paper, we prove that any two dimensional complete translating soliton in ${\mathbb R}^4$ with finite total curvature is a plane. It is in fact a consequence of our main result.
In this article, we establish pinching conditions under which all weakly stable Yang-Mills connections on compact manifolds are flat. As a corollary, we provide a dimension-dependent constant $\delta(n)$ and prove that there exist no non-flat weakly stable Yang-Mills connections on $\delta(n)$-pinched compact simply-connected Riemannian manifolds.
In this paper we prove that there is a neighborhood in the $C^2$ topology of the usual metric on the Euclidean sphere $S^n (n\geq 5)$ such that there is no nontrivial weakly stable Yang-Mills connections for any metric $\tilde{g}$ in this neighborhood. We also study the stability of Yang-Mills connections on the warped product manifolds.
It is known that there is no a Type I singularity for the Lagrangian mean curvature flow with zero Maslov class. In this paper, we study translating solitons which are important models of Type II singularities. A necessary condition for a blow-up limit arising at a Type II singularity of a Lagrangian mean curvature flow with zero Maslov class is provided. As an application, we try to understand the important open question proposed by Joyce-Lee-Tsui and Neves-Tian, whether the Lagrangian translating solitons constructed by Joyce-Lee-Tsui can be a blow-up limit for a Lagrangian mean curvature flow with zero Maslov class.
In this paper, we study the properties of the critical points of Yang-Mills-Higgs functional, which are called Yang-Mills-Higgs pairs. We first consider the properties of weakly stable Yang-Mills-Higgs pairs on a vector bundle over S^n (n > 3). When n > 3, we prove that the norm of its Higgs field is 1 and the connection is actually Yang-Mills. More precisely, its curvature vanishes when n > 4. We also use the bubble-neck decomposition to prove the energy identity of a sequence of Yang-Mills-Higgs pairs over a 4-dimensional compact manifold with uniformly bounded energy. We show there is a subsequence converges smoothly to a Yang-Mills-Higgs pair up to gauge modulo finitely many 4-dimensional spheres with Yang-Mills connections.
In this paper, we investigate the Chern number inequalities on $4$-dimensional K\"ahler manifolds admitting the deformed Hermitian-Yang-Mills metrics under the assumption $\hat\theta\in (\pi,2\pi)$.
Let $(X,\omega)$ be a compact K\"ahler manifold of complex dimension $n$ and $(L,h)$ be a holomorphic line bundle over $X$. The line bundle mean curvature flow was introduced in \cite{JY} in order to find deformed Hermitian-Yang-Mills metrics on $L$. In this paper, we consider the stability of the line bundle mean curvature flow. Suppose there exists a deformed Hermitian Yang-Mills metric $\hat h$ on $L$. We prove that the line bundle mean curvature flow converges to $\hat h$ exponentially in $C^\infty$ sense as long as the initial metric is close to $\hat h$ in $C^2$-norm.
In this paper, we study the deformed Hermitian-Yang-Mills equation on compact K\"ahler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric if there exists a positive constant $C$ such that $-\frac{1}{C}\omega<\sqrt{-1}F<C\omega$. We also study the self-shrinker over $\mathbb{C}^n$ to the corresponding parabolic flow. We prove that the self-shrinker over $\mathbb{C}^n$ is a quadratic polynomial function. We also show the similar rigid theorem for the J-equations and the self-shrinkers over $\mathbb{C}^n$ to J-flow.
In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equation $$ \Delta^{2} u -\Delta u+(\lambda a+1)u= |u|^{p-2}u $$ on $G=(V,E)$. Under some suitable assumptions, we prove that for any $\lambda>1$ and $p>2$, the equation admits a ground state solution $u_{\lambda}$. Moreover, we prove that as $\lambda\rightarrow +\infty$, the solutions $u_{\lambda}$ converge to a solution of the equation \begin{align*} \begin{cases} \Delta^{2}u -\Delta u+u = |u|^{p-2}u, &\text{in}\ \ \Omega, u=0, &\text{on}\ \ \partial\Omega, \end{cases} \end{align*} where $\Omega=\{x\in V: a(x)=0\}$ is the potential well and $\partial\Omega$ denotes the the boundary of $\Omega$.
In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given K\"ahler manifold. The goal of this paper is to give an $\varepsilon$-regularity theorem for the line bundle mean curvature flow. To establish the theorem, we provide a scale invariant monotone quantity. As a critical point of this quantity, we define self-shrinker solution of the line bundle mean curvature flow. The Liouville type theorem for self-shrinkers is also given. It plays an important role in the proof of the $\varepsilon$-regularity theorem.
We investigate a parabolic-elliptic system for maps $(u,v)$ from a compact Riemann surface $M$ into a Lorentzian manifold $N\times{\mathbb{R}}$ with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the $v$-equation as a nonlinear second order constraint along the flow. We prove a global existence result of the parabolic-elliptic system by assuming either some geometric conditions on the target Lorentzian manifold or small energy of the initial maps. The result implies the existence of a Lorentzian harmonic map in a given homotopy class with fixed boundary data.
We investigate a parabolic-elliptic system which is related to a harmonic map from a compact Riemann surface with a smooth boundary into a Lorentzian manifold with a warped product metric. We prove that there exists a unique global weak solution for this system which is regular except for at most finitely many singular points.
In this paper we derive the Euler-Lagrange equation of the functional $L_\beta=\int_\Sigma\frac{1}{\cos^\beta\alpha}d\mu, ~~\beta\neq -1$ in the class of symplectic surfaces. It is $\cos^3\alpha {\bf{H}}=\beta(J(J\nabla\cos\alpha)^\top)^\bot$, which is an elliptic equation when $\beta\geq 0$. We call such a surface a $\beta$-symplectic critical surface. We first study the properties for each fixed $\beta$-symplectic critical surface and then prove that the set of $\beta$ where there is a stable $\beta$-symplectic critical surface is open. We believe it should be also closed. As a precise example, we study rotationally symmetric $\beta$-symplectic critical surfaces in ${\mathbb C}^2$ carefully .
We prove a non-collapsing property for curvature flows of embedded hypersurfaces in the sphere and in hyperbolic space.
Let $(M,\bar{g})$ be a Kähler surface with a constant holomorphic sectional curvature $k>0$, and $Σ$ an immersed symplectic surface in $M$. Suppose $Σ$ evolves along the mean curvature flow in $M$. In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies $|A|^2\leq 2/3|H|^2+1/2 k$ and $\cosα\geq \frac{\sqrt{30}}{6}$ or $|A|^2\leq 2/3 |H|^2+4/5 k\cosα$ and $\cosα\ge251/265$.
Let $(M,\overline{g})$ be a Kähler surface, and $Σ$ an immersed surface in $M$. The Kähler angle of $Σ$ in $M$ is introduced by Chern-Wolfson \cite{CW}. Let $(M,\overline{g}(t))$ evolve along the Kähler-Ricci flow, and $Σ_t$ in $(M,\overline{g}(t))$ evolve along the mean curvature flow. We show that the Kähler angle $α(t)$ satisfies the evolution equation: $$ (\frac{\partial}{\partial t}-Δ)\cosα=|\overline\nabla J_{Σ_t}|^2\cosα+R\sin^2α\cosα, $$ where $R$ is the scalar curvature of $(M, \overline{g}(t))$. The equation implies that, if the initial surface is symplectic (Lagrangian), then along the flow, $Σ_t$ is always symplectic (Lagrangian) at each time $t$, which we call a symplectic (Lagrangian) Kähler-Ricci mean curvature flow. In this paper, we mainly study the symplectic Kähler-Ricci mean curvature flow.
In this paper, we will derive a small energy regularity theorem for the mean curvature flow of arbitrary dimension and codimension. It says that if the parabolic integral of $|A|^2$ around a point in space-time is small, then the mean curvature flow cannot develop singularity at this point. As an application, we can prove that the 2-dimensional Hausdorff measure of the singular set of the mean curvature flow from a surface to a Riemannian manifold must be zero.
In this paper we study the singularities of the mean curvature flow from a symplectic surface or from a Lagrangian surface in a Kähler-Einstein surface. We prove that the blow-up flow $Σ_s^\infty$ at a singular point $(X_0, T_0)$ of a symplectic mean curvature flow $Σ_t$ or of a Lagrangian mean curvature flow $Σ_t$ is a non trivial minimal surface in ${\bf R}^4$, if $Σ_{-\infty}^\infty$ is connected.