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# 数学代写|黎曼曲面代写Riemann surface代考|Bloch domains

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## 数学代写|黎曼曲面代写Riemann surface代考|Bloch domains

We have seen that the Poincaré distance in an hyperbolic Riemann surface $X$ is necessarily complete; in particular, for every $z \in X$ and $r>0$ the Poincaré ball $B_X(z, r)$ is relatively compact in $X$. On the other hand, if $\Omega$ is a subdomain of $X$ it might well happen that there exists a radius $r>0$ such that all Poincaré balls with respect to the Poincaré distance of $X$ of radius $r$ centered in a point of $\Omega$ intersect $\partial \Omega$. Such subdomains will be used in Chapter 3 in our discussion of random iteration; this section is devoted to study and characterize them.

Definition 1.11.1. If $\Omega \subset X$ is a domain in a hyperbolic Riemann surface $X$ and $z \in \Omega$ put

$$R(z ; \Omega, X)=\sup \left{r>0 \mid B_X(z, r) \subseteq \Omega\right}$$
and
$$R(\Omega, X)=\sup _{z \in \Omega} R(z ; \Omega, X)=\sup \left{r>0 \mid \exists z \in \Omega: B_X(z, r) \subseteq \Omega\right}$$
We say that $\Omega$ is a Bloch domain of $X$ if $R(\Omega, X)<+\infty$.
It is not too difficult to find examples of Bloch domains and of domains that are not Bloch.

## 数学代写|黎曼曲面代写Riemann surface代考|Boundary Schwarz lemmas

In the first chapter we learned to appreciate the importance of the Schwarz lemma. Unfortunately, its original form has a shortcoming: it cannot be directly applied to get information about boundary behaviors. Julia first and Wolff shortly later overcame this difficulty, proving the lemmas known under their names, which are the main topic of this chapter.

Their idea is simple. The Schwarz-Pick lemma says that a holomorphic function $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ sends Poincaré balls into Poincaré balls. Then to get information about a boundary point $\tau \in \partial \mathbb{D}$ one can choose a sequence of Poincaré balls with centers converging to $\tau$ and constant Euclidean radius, apply the Schwarz-Pick lemma to each one of them, and take the limit. In particular, it turns out that the right geometrical objects to consider are the horocycles: Euclidean disks internally tangent to a point of $\partial \mathbb{D}$. In fact, the Julia and Wolff lemmas say that, under suitable hypotheses, a holomorphic function $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ sends horocycles into horocycles in a very controlled way.
In this chapter, we shall also discuss some applications of the Julia and Wolff lemmas, leaving the main one-dynamics-to the next chapters. The first application concerns the behavior of the angular derivative. Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$, take $\sigma \in \partial \mathbb{D}$, and assume for the sake of simplicity, that $f(z) \rightarrow \tau \in \partial \mathbb{D}$ as $z \rightarrow \sigma$. Then we would like to know something about the behavior of the derivative $f^{\prime}$ near $\sigma$. A natural approach is to study the incremental ratio $(f(z)-\tau) /(z-\sigma)$, and the Julia lemma turns out to be the ideal tool for this investigation. We shall see that a suitable limit, the nontangential limit, of the incremental ratio at $\sigma$ exists, possibly equal to infinity; moreover, if it is finite, it coincides with the nontangential limit of the derivative at $\sigma$. This will also allow us to give a criterion for the existence of the nontangential limit of $f^{\prime}$ at a boundary point.

## 数学代写|黎曼曲面代写Riemann surface代考|Bloch domains

1.11.1.定义如果$\Omega \subset X$是双曲黎曼曲面上的一个定义域$X$和$z \in \Omega$

$$R(z ; \Omega, X)=\sup \left{r>0 \mid B_X(z, r) \subseteq \Omega\right}$$

$$R(\Omega, X)=\sup _{z \in \Omega} R(z ; \Omega, X)=\sup \left{r>0 \mid \exists z \in \Omega: B_X(z, r) \subseteq \Omega\right}$$

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