标度演算中的反例
Counterexamples in Scale Calculus
摘要：
我们构造反例经典微积分的事实，如逆和隐函数定理的规模演算-一个推广的多变量微积分无穷维向量空间中的重参数化映射相关的辛几何是光滑的。标度演算是多重折叠理论的基石，多重折叠理论是由Hofer-Wysocki-Zehnder引入的，作为一种广泛适用的工具来正则化伪全纯曲线的模空间。我们展示了新的非线性尺度Fredholm概念在多重折叠理论克服了缺乏隐函数定理，正式建立一个经常隐含使用的事实：基本芽的微分-局部模型的尺度Fredholm映射-连续变化的空间中的有界算子时，基点的变化。此外，我们证明，这种连续性只在特定的坐标，通过构建一个例子的规模-非同态和规模-Fredholm映射的不连续微分。这证明了多重折叠理论基础的高技术复杂性。
英文摘要：
We construct counterexamples to classical calculus facts such as the Inverse and Implicit Function Theorems in Scale Calculus -- a generalization of Multivariable Calculus to infinite dimensional vector spaces in which the reparameterization maps relevant to Symplectic Geometry are smooth. Scale Calculus is a cornerstone of Polyfold Theory, which was introduced by Hofer-Wysocki-Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves. We show how the novel nonlinear scale-Fredholm notion in Polyfold Theory overcomes the lack of Implicit Function Theorems, by formally establishing an often implicitly used fact: The differentials of basic germs -- the local models for scale-Fredholm maps -- vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of Polyfold Theory.
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