The blow-up analysis of an affine Toda system corresponding to superconformal minimal surfaces in S-4

摘要

In this paper, we study the blow-up analysis of an affine Toda system corresponding to minimal surfaces into S-4. This system is an integrable system which is a natural generalization of sinh-Gordon equation. By exploring a refined blow-up analysis in the bubble domain, we prove that the blow-up values are multiple of 8p, which generalizes the previous results proved in [39,37,27,22] for the sinh-Gordon equation. More precisely, let (u(k)(1), u(k)(2), u(k)(3)) be a sequence of solutions of -Delta u(1) = e(u1) - e(u3), -Delta u(2) = e(u2) - e(u3), -Delta u(3) = - 1/2 e(u1) - 1/2 e(u2) + e(u3), u(1) + u(2) + 2u(3) = 0, in B-1(0), which has a uniformly bounded energy in B-1(0), a uniformly bounded oscillation on partial derivative B-1(0) and blows up at an isolated blow-up point {0}, then the local masses (sigma(1), sigma(2), sigma(3)) not equal 0 satisfy sigma(1) = m(1)( m(1) + 3)+ m(2)( m(2) - 1) sigma(2) = m(1)( m(1) - 1) + m(2)( m(2) + 3) sigma(3) = m(1)( m(1) - 1) + m(2)( m(2) - 1) is an element of Z either for some m1, m2 = 0, 1 mod4, or m1, m2 = 2, 3 mod4. Here sigma(i):= 1/2 pi lim(delta -> 0) lim(k ->infinity) integral(Bd)(0)e(u)i(k)dx. (C) 2021 Elsevier Inc. All rights reserved.