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 $\lim _{k\to \infty }d(u_k^*\mathcal {J}^\alpha )$, for $\alpha =1,2,3$, in the weak sense. As a corollary, we show that the maps constructed by Chen-Li [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), pp. 375–388] and by Foscolo [J. Differential Geom. 112 (2019), pp. 79–120] cannot be tangent maps (c.f Li and Tian [Internat. Math. Res. Notices 14 (1998), pp. 735–755], Theorem 3.1) of a stationary quaternionic map satisfing $d(u^*\mathcal {J}^\alpha )=0$.