### Hitchin' s self-duality equations on complete Riemannian manifolds.

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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 $(N,{\mathcal{J}}^{\alpha},\alpha =1,2,3)$ 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 $({\mathbb{R}}^{m},{e}^{{\left|x\right|}^{2}/2(m-2)}/d{s}_{0}^{2})$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds{2}_{0}$ is the Euclidean metric in ${\mathbb{R}}^{m}$. 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...

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