A blow-up formula for stationary quaternionic maps

Abstract

Let $(M,J^{\alpha },\alpha =1,2,3)$ and $(N,{\mathcal{J}}^{\alpha },\alpha =1,2,3)$ be Hyperkähler manifolds. Suppose that $u_k$ is a sequence of stationary quaternionic maps and converges weakly to $u$ in $H^{1,2}(M,N)$, we derive a blow-up formula for ${\lim }_{k\to \infty }d(u_k^{\ast }{\mathcal{J}}^{\alpha })$, for $\alpha =1,2,3$, in the weak sense. As a corollary, we show that the maps constructed by Chen-Li [CL2] and by Foscolo [F] can not be tangent maps (c.f [LT], Theorem 3.1) of a stationary quaternionic map satisfing $d(u^{\ast }{\mathcal{J}}^{\alpha })=0$.

7 pages