A blow-up formula for stationary quaternionic maps

Abstract

Let ( M , J α , α = 1 , 2 , 3 ) (M, J^\alpha , \alpha =1,2,3) and ( N , J α , α = 1 , 2 , 3 ) (N, \mathcal {J}^\alpha , \alpha =1,2,3) be Hyperkähler manifolds. Suppose that u k u_k is a sequence of stationary quaternionic maps and converges weakly to u u in H 1 , 2 ( M , N ) H^{1,2}(M,N) , we derive a blow-up formula for lim k → ∞ d ( u k ∗ J α ) \lim _{k\to \infty }d(u_k^*\mathcal {J}^\alpha ) , for α = 1 , 2 , 3 \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 ∗ J α ) = 0 d(u^*\mathcal {J}^\alpha )=0 .