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The gradient flow of Higgs pairs

Jiayu LiXi Zhang — 2011

Journal of the European Mathematical Society

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 , ω ) , 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 , φ 0 ) converges, in an appropriate sense which takes into account bubbling phenomena, to a critical point ( A , φ ) of this functional. We also prove that the limiting Higgs pair ( A , φ ) can be extended smoothly to a vector bundle E over...

Liouville theorems for self-similar solutions of heat flows

Jiayu LiMeng Wang — 2009

Journal of the European Mathematical Society

Let N be a compact Riemannian manifold. A quasi-harmonic sphere on N is a harmonic map from ( m , e | x | 2 / 2 ( m - 2 ) / d s 0 2 ) to N ( m 3 ) with finite energy ([LnW]). Here d s 2 0 is the Euclidean metric in 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...

Symplectic critical surfaces in Kähler surfaces

Xiaoli HanJiayu Li — 2010

Journal of the European Mathematical Society

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 = Σ 1 cos α d μ in the class of symplectic surfaces. It is cos 3 α H = ( J ( J cos α ) ) , 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 call such a surface a symplectic critical surface. We show that, if M is a Kähler-Einstein surface with nonnegative...

Quaternionic maps and minimal surfaces

Jingyi ChenJiayu Li — 2005

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let ( M , J α , α = 1 , 2 , 3 ) and ( N , 𝒥 α , α = 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...

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