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In this paper we study the solutions of Toda systems on Riemann surface in the critical case, proving a sufficient condition for existence.

Let $(M,J^{\alpha },\alpha =1,2,3)$ and $(N,{𝒥}^{\alpha },\alpha =1,2,3)$ be hyperkähler manifolds. We study stationary quaternionic maps between $M$ and $N$. We first show that if there are no holomorphic 2-spheres in the target then any sequence of stationary quaternionic maps with bounded energy subconverges to a stationary quaternionic map strongly in $W^{1,2}(M,N)$. We then find that certain tangent maps of quaternionic maps give rise to an interesting minimal 2-sphere. At last we construct a stationary quaternionic map with a codimension-3 singular set by using the embedded...

Let $M$ be a Kähler surface and $\Sigma$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the Kähler angle of $\Sigma$ in $M$. We first deduce the Euler-Lagrange equation of the functional $L={\int }_{\Sigma }\frac{1}{cos\alpha }d\mu$ in the class of symplectic surfaces. It is ${cos}^3\alpha H={\left(J{(J\nabla cos\alpha )}^{\top }\right)}^{\perp }$, where $H$ is the mean curvature vector of $\Sigma$ in $M$, $J$ is the complex structure compatible with the Kähler form $\omega$ in $M$, which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface with nonnegative...

We consider the gradient flow of the Yang–Mills–Higgs functional of Higgs pairs on a Hermitian vector bundle $(E,H_0)$ over a Kähler surface $(M,\omega )$, and study the asymptotic behavior of the heat flow for Higgs pairs at infinity. The main result is that the gradient flow with initial condition $(A_0,{\phi }_0)$ converges, in an appropriate sense which takes into account bubbling phenomena, to a critical point $(A_{\infty },{\phi }_{\infty })$ of this functional. We also prove that the limiting Higgs pair $(A_{\infty },{\phi }_{\infty })$ can be extended smoothly to a vector bundle $E_{\infty }$ over...

Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $({ℝ}^m,e^{{\left|x\right|}^2/2(m-2)}/d{s}_0^2)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds{2}_0$ is the Euclidean metric in ${ℝ}^m$. Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive...