论文标题
Lagrange乘数和绝热限制
Lagrange multipliers and adiabatic limits I
论文作者
论文摘要
可以通过将功能限制到约束或查找Lagrange乘数功能的关键点来检测到受约束的函数的关键点。尽管两个功能的临界点，即限制和拉格朗日乘数功能在自然的一对一对应中，这对于它们的梯度流线不需要正确。我们考虑了度量标准的奇异变形，并通过绝热的极限论证显示，接近奇异性，我们在连接Morse索引差差的临界点的梯度流线之间具有一对一的对应关系。我们在[FW22B]文章中介绍了绝热极限技术的一般概述。 信件的证明分为两部分。当前的部分我处理的线性方法导致隐式函数定理的单数版本。我们还讨论了Rabinowitz-loer同源性中可能的无限尺寸概括。在第二部分[FW22A]中，我们采用非线性方法，特别是证明了紧凑的结果和均匀的指数衰减，而与变形参数无关。
Critical points of a function subject to a constraint can be either detected by restricting the function to the constraint or by looking for critical points of the Lagrange multiplier functional. Although the critical points of the two functionals, namely the restriction and the Lagrange multiplier functional are in natural one-to-one correspondence this does not need to be true for their gradient flow lines. We consider a singular deformation of the metric and show by an adiabatic limit argument that close to the singularity we have a one-to-one correspondence between gradient flow lines connecting critical points of Morse index difference one. We present a general overview of the adiabatic limit technique in the article [FW22b]. The proof of the correspondence is carried out in two parts. The current part I deals with linear methods leading to a singular version of the implicit function theorem. We also discuss possible infinite dimensional generalizations in Rabinowitz-Floer homology. In part II [FW22a] we apply non-linear methods and prove, in particular, a compactness result and uniform exponential decay independent of the deformation parameter.
