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# 复分析代考_Complex analysis代考_MATH3401 Isolated fixed point sets, cardinality

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## 复分析代考_Complex analysis代考_Isolated fixed point sets, cardinality

In classical mechanics the following Euler’s theorem is well known: the general displacement of a rigid body with one point fixed is a rotation about some axis. So, if one considers an orientation-preserving isometry of a domain in $\mathbb{R}^3$ fixing one point, the fixed point set of this isometry will necessarily contain at least a segment, so the fixed point set cannot be a discrete set. In the euclidean space $\mathbb{R}^n$, one can always find a domain which has a euclidean isometry with exactly one fixed point, however for any $n$, if an isometry of a domain in $\mathbb{R}^n$ has two fixed points it will force the existence of at least a segment to belong to the fixed point set, and so this set will be at least one dimensional.

Switching to complex analysis, we remark that any holomorphic automorphism of a bounded domain in $\mathbb{C}^n$ (or in general, hyperbolic manifold) is an isometry in an invariant metric, so an Euler type statement is certainly meaningful, that is if this automorphism has a discrete fixed point set one can inquire what its cardinality and structure might be. To describe this more precisely, let $f: M \rightarrow M$ be a holomorphic self-map of a complex manifold M. Suppose that $\operatorname{Fix}(f)$ is discrete. We shall examine mostly two questions. First, how large this set can be for specific cases: $M$ is a bounded domain in $\mathbb{C}^n$, a hyperbolic manifold, etc., while $f$ is a holomorphic automorphism or endomorphism. Second, the structure of $\operatorname{Fix}(f)$, namely which points of $M$ could form such a set for some holomorphic self-map of $M$. Everywhere below we consider only holomorphic self-maps (automorphisms or endomorphisms) of various complex manifolds, and for the sake of compactness the word holomorphic may be omitted.

## 复分析代考_Complex analysis代考_Fixed point sets consisting of one or two points

In this section we’ll discuss which subsets of a manifold $D$ can be $F i x(f)$ for some holomorphic automorphism or endomorphism $f: D \rightarrow D$. As the title suggests we’ll consider two cases.

First we consider the case when every single point of a domain is the Fix $(f)$ for a suitable holomorphic $f$.

Theorem 4.6.10. [28, Theorem 2.1] If every point of a hyperbolic manifold $D$ is a fixed point set for some holomorphic automorphism of $D$, then $D$ is a homogeneous manifold.

Proof. 1. First we note that the theorem will follow from a local statement: let $x \in D$, then there exists a neighborhood $U_x$ of $x$ such that for any $y \in U_x$ there is a $g \in \operatorname{Aut}(D)$ such that $g(y)=x$. Indeed, if this is true consider two arbitrary points $a, b \in D$, connect them by a compact path $L$, cover $L$ by a finite number of $U_x, x \in L$, and one can obtain an $f \in \operatorname{Aut}(D)$, such that $f(a)=b$.

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