论文标题
对应于$ {\ mathbb s}^4 $的超符号最小表面的仿射TODA系统的爆破分析
The blow-up analysis of an affine Toda system corresponding to superconformal minimal surfaces in ${\mathbb S}^4$
论文作者
论文摘要
在本文中，我们研究了对应于最小表面的仿射TODA系统的爆炸分析，以$ {\ Mathbb s}^4 $ [19]。该系统是一个可集成的系统，是SINH-GORDON方程的自然概括[18]。通过探索气泡域中的精制爆破分析，我们证明了爆炸值的$8π$，它概括了先前的结果，以\ cite {spruck，os，jost-wang-ye-zhou，jevnikar-wei-yang}的范围，用于Sinh-gordon方程。令$（u_k^1，u_k^2，u^3_k）$为\ begin {align*}-Δu^1＆= e^{u^1} -e^{u^{u^3}，\\\\\\-ΔU^2＆= e^2＆= e^2} { -ΔU^3＆= - \ \ frac {1} {2} e^{u^1} - \ frac {1} {1} {2} e^{u^2}+e^{u^{u^3}，\\ \ \ \ u^1+u^1+u^2+2U^3＆= 0，\ eend end in of en of en of in osign in osign of $ b_1（0）$，$ \ $ \ a部分b_1（0）$的均匀边界振荡，并在一个孤立的爆炸点$ \ {0 \} $上吹来，然后本地质量$（σ_1，σ_2，σ_3） σ_1＆=＆m_1（m_1+3）+m_2（m_2-1）\\σ_2＆=＆m_1（m_1-1）+m_2（m_2+3）\\σ_3＆=＆m_1（m_1-1）（m_1-1）+m_2（m_2-1） \ begin {array} {l}（m_1，m_2）\ in {\ mathbb z} \ hbox {with} \\ m_1，m_2 = 0 \ hbox {or} {or} 1 \ hbox {mod} {mod} 4，\\ m_1，m_1，m_2 = 2 = 2 = 2 \ hbox \ hbox \ hbox} 4 bbox {or} 4 bbox {or} {or} {或} {或} {或} {或} 3 \ {\ {4 bbox {or} \ end {array} \ end {align*}这里局部质量由$σ_i：= \ frac 1 {2π}} \ lim_ {Δ\ to 0} \ lim_ {k \ to \ infty} \ infty} \ int_ \ int_
In this paper, we study the blow-up analysis of an affine Toda system corresponding to minimal surfaces into ${\mathbb S}^4$ [19]. This system is an integrable system which is a natural generalization of sinh-Gordon equation [18]. By exploring a refined blow-up analysis in the bubble domain, we prove that the blow-up values are multiple of $8π$, which generalizes the previous results proved in \cite{Spruck, OS, Jost-Wang-Ye-Zhou, Jevnikar-Wei-Yang} for the sinh-Gordon equation. Let $(u_k^1,u_k^2, u^3_k)$ be a sequence of solutions of \begin{align*} -Δu^1&=e^{u^1}-e^{u^3},\\ -Δu^2&=e^{u^2}-e^{u^3},\\ -Δu^3&=-\frac{1}{2}e^{u^1}-\frac{1}{2}e^{u^2}+ e^{u^3},\\ u^1+u^2+2u^3&=0, \end{align*} in $B_1(0)$, which has a uniformly bounded energy in $B_1(0)$, a uniformly bounded oscillation on $\partial B_1(0)$ and blows up at an isolated blow-up point $\{0\}$, then the local masses $(σ_1,σ_2, σ_3) \not = 0$ satisfy \begin{align*} \begin{array}{rcl} σ_1&=&m_1(m_1+3)+m_2(m_2-1)\\ σ_2&=& m_1(m_1-1)+m_2(m_2+3)\\ σ_3 &=& m_1(m_1-1)+m_2(m_2-1) \end{array} \, \qquad \hbox { for some } \begin{array} {l} (m_1, m_2)\in {\mathbb Z} \hbox { with }\\ m_1, m_2= 0 \hbox{ or } 1 \hbox{ mod } 4,\\ m_1, m_2 = 2 \hbox { or } 3\hbox { mod } 4. \end{array} \end{align*} Here the local mass is defined by $ σ_i:=\frac 1{2π}\lim_{δ\to 0}\lim_{k\to\infty}\int_{B_δ(0)}e^{u_k^i}dx.$
