U(1) -invariant special Lagrangian 3-folds. I. Nonsingular solutions
U(1) -invariant special Lagrangian 3-folds. I. Nonsingular solutions
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U(1) - 不变的特殊拉格朗日三重。
DOI：
10.1016/j.aim.2004.04.001
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发表时间：
2001
影响因子：
1.7
通讯作者：
D. joyce
中科院分区：
文献类型：
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作者：
D. joyce
本文用解析的方法研究了在U(1)-作用Eiθ：(Z1，Z2，Z3)↦(EiθZ1，e−IθZ2，Z3)作用下C3不变量中的特殊拉格朗日3-子流形(SL 3-折叠)N。设N是这样的U(1)-不变SL 3重。则|Z1|2−|Z2|2=2a在N上，对于某个a∈R。在局部上，N可以表示为满足依赖于a的非线性柯西-黎曼方程的函数u，v：R2→R的一种图形，使得u+iv类似于x+iy的全纯函数。当a非零时，u，v总是光滑的，N总是非奇异的。但如果a=0，则可能存在u、v不可微的点(x，0)，这些点对应于N的奇点。本文主要讨论当a非零时的非奇异情形。我们证明了复变分析中著名结果的非线性柯西-黎曼方程的类似性。特别地，我们证明了由它衍生的两个Dirichlet问题解的存在唯一性。这证明了C3中一大类非奇异U(1)不变SL3-折叠在两种边界条件下的存在唯一性。把这些推广到了a=0的情形，研究了由此产生的SL 3-折叠的奇性，并构造了C3中开集的特殊的Lagrangian性质。
This is the first of three papers studying special Lagrangian 3-submanifolds (SL 3-folds) N in C3invariant under the U(1)-action eiθ:(z1,z2,z3)↦(eiθz1,e−iθz2,z3), using analytic methods. Let N be such a U(1)-invariant SL 3-fold. Then |z1|2−|z2|2=2a on N for some a∈ R . Locally, N can be written as a kind of graph of functions u,v: R2→ R satisfying a nonlinear Cauchy–Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy. When a is nonzero, u,v are always smooth and N is always nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, which correspond to singular points of N. This paper focusses on the nonsingular case, when a is nonzero. We prove analogues for our nonlinear Cauchy–Riemann equation of well-known results in complex analysis. In particular, we prove existence and uniqueness for solutions of two Dirichlet problems derived from it. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C3, with two kinds of boundary conditions. The sequels extend these to the case a=0, study the singularities of the SL 3-folds that arise, and construct special Lagrangian fibrations of open sets in C3.