A blow-up formula for stationary quaternionic maps

TL;DR

This paper derives a blow-up formula for stationary quaternionic maps between Hyperk"ahler manifolds, revealing limitations on tangent maps and advancing understanding of their weak convergence behavior.

Contribution

It introduces a blow-up formula for stationary quaternionic maps and shows certain constructed maps cannot be tangent maps of such stationary maps.

Findings

Derived a blow-up formula for weak limits of quaternionic maps

Proved that specific constructed maps are not tangent maps of stationary quaternionic maps

Enhanced understanding of the weak convergence and singularity formation in quaternionic harmonic maps

Abstract

Let $(M, J^\alpha, \alpha =1,2,3)$ and $(N, {\cal J}^\alpha, \alpha =1,2,3)$ be Hyperk\"ahler 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 $\lim_{k\to\infty}d(u_k^*{\cal J}^\alpha)$, 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 $d(u^*{\cal J}^\alpha)=0$.