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Jiayu Li

Publications and source records attributed to Jiayu Li.

159 records · Page 9Linked to original sources
arXiv2011

The mean curvature flow along the Kähler-Ricci flow

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.

math.DG↗
arXiv2010

Nonexistence of quasi-harmonic sphere with large energy

Nonexistence of quasi-harmonic spheres is necessary for long time existence and convergence of harmonic map heat flows. Let $(N,h)$ be a complete noncompact Riemannian manifolds. Assume the universal covering of $(N,h)$ admits a nonnegative strictly convex function with polynomial growth. Then there is no quasi-harmonic spheres $u:\mathbb{R}^n\ra N$ such that $$\lim_{r\ra\infty}r^ne^{-\f{r^2}{4}}\int_{|x|\leq r}e^{-\f{|x|^2}{4}}|\nabla u|^2dx=0.$$ This generalizes a result of the first named author and X. Zhu (Calc. Var., 2009). Our method is essentially the Moser iteration and thus very simple.

math.DG↗
arXiv2008

Singularities of symplectic and Lagrangian mean curvature flows

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.

math.DG↗
arXiv2008

Translating solitons to symplectic and Lagrangian mean curvature flows

In this paper, we construct finite blow-up examples for symplectic mean curvature flows and we study properties of symplectic translating solitons. We prove that, the Kähler angle $α$ of a symplectic translating soliton with $\max |A|=1$ satisfies that $\sup |α|>\fracπ{4}\frac{|T|}{|T|+1}$ where $T$ is the direction in which the surface transltes.

math.DG↗
arXiv2007

Symplectic critical surfaces in Kähler surfaces

Let $M$ be a Kähler surface and $Σ$ be a closed symplectic surface which is smoothly immersed in $M$. Let $α$ be the Kähler angle of $Σ$ in $M$. We first deduce the Euler-Lagrange equation of the functional $L=\int_Σ\frac{1}{\cosα}dμ$ in the class of symplectic surfaces. It is $\cos^3αH=(J(J\nabla\cosα)^\top)^\bot$, where $H$ is the mean curvature vector of $Σ$ in $M$, $J$ is the complex structure compatible with the Kähler form $ω$ in $M$, which is an elliptic equation. We then study the properties of the equation.

math.DG↗
arXiv2004

Dirac-Harmonic Maps

We introduce a functional that couples the nonlinear sigma model with a spinor field: $L=\int_M[|dϕ|^2+(ψ,\Dψ)]$. In two dimensions, it is conformally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a removable singularity theorem.

math.DG↗
arXiv2004

Regularity Theorems and Energy Identities for Dirac-Harmonic Maps

We study a new set of coupled field equations motivated by the non-linear supersymmetric sigma model of quantum field theory. These equations couple a map into a Riemannian manifold controlled by a harmonic map like action with a spinor field along that map. We study the solutions which we call Dirac-harmonic maps from a Riemann surface to a sphere $§^n$. We show that a weakly Dirac-harmonic map is in fact smooth, and prove that the energy identity holds during the blow-up process.

math.DG↗
arXiv2003

Harmonic bundles on quasi-compact Kaehler manifolds

We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that any discrete sub group of a Lie group of rank >1 that is not of Hodge type can't be the fundamental group of a quasi-compact Kaehler manifold.

math.AG↗
arXiv2003

Singularity of Mean Curvature Flow of Lagrangian Submanifolds

In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold is Lagrangian and almost calibrated by ReΩin a Calabi-Yau n-fold (M,Ω), and T>0 is the first blow-up time of the mean curvature flow, then the tangent cone of the mean curvature flow at a singular point (X,T) is a stationary Lagrangian integer multiplicity current in R\sup 2n with volume density greater than one at X. When n=2, the tangent cone consists of a finite union of more than one 2-planes in R\sup 4 which are complex in a complex structure on R\sup 4.

math.DG↗
arXiv2001

Harmonic Hopf Constructions Between Spheres II

We prove that a necessary condition for the existence of the remaining problem in the harmonic Hopf construction is also sufficient. We also give some topological applications based on our result.

math.DG↗