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\begin{document}

\centerline{\Large Holomorphic maps between Riemann
  surfaces}
\vskip 0.5cm
\centerline{\Large and Normal Families}

\vskip 1cm



\subsection*{Holomorphic maps between tori and other Riemann surfaces.}

\begin{enumerate}[1.]
\item Construct a surjective holomorphic map $\mathbb D\to\C$.
  \item Recall annuli $A(D)=\{D<|z|<1\}$ for $0\leq D<1$. For which
    $D$ is there a holomorphic map $f:A(D)\to A(D)$ of degree 2,
    meaning that the induced homomorphism
    $f_*:\pi_1(A(D))\to\pi_1(A(D))$ is multiplication by 2? 
  \item
  Let $T_\Lambda=\C/\Lambda$ and $T_\Gamma=\C/\Gamma$ be two tori with
  $\Lambda,\Gamma\subset\C$ two lattices. Prove that every holomorphic
  map $f:T_\Lambda\to T_\Gamma$ is either constant or a covering
  map. Hint: Consider two cases. If the image of
  $f_*:\pi_1(\Lambda)\to\pi_1(\Gamma)$ has infinite index, show that
  $f$ is constant by lifting to suitable covering spaces so that you
  get a map from a compact to a noncompact Riemann surface. If the
  image has finite index, show that the lift $\tilde f:\C\to\C$
  between universal covers grows linearly: $|\tilde f(z)|\leq A|z|+B$
  for some constants $A,B$, and conclude that $\tilde f$ is a linear
  function.
  \item Let $\Lambda=\Z[i]$ be the Gaussian integers, and
    $\Gamma=\Z[\omega]$ be the Eisenstein integers, where
    $\omega=e^{\pi i/3}$. Show that every holomorphic map
    $T_\Lambda\to T_\Gamma$ and $T_\Gamma\to T_\Lambda$ is constant.
    Hint: This should reduce to showing that there is no $a\neq 0$ so
    that $a\Lambda\subset \Gamma$ or $a\Gamma\subset
    \Lambda$. Specifically, show that $\omega\not\in\Q(i)$ and
    $i\not\in \Q(\omega)$ (fields obtained by adjoining a number).

  \item Recall that the Schwarz lemma says that a holomorphic map
    $f:\D\to\D$ is either a hyperbolic isometry or a strict
    contraction: $\|df(v)\|<\|v\|$ for all nonzero tangent vectors
    $v$, where $\|\cdot\|$ is hyperbolic norm. Show that this
    cannot be improved to the statement: $f$ is either an isometry or
    there is some $c<1$ such that $\|df(v)\|\leq c\|v\|$. Hint:
    $z\mapsto z^2$.

 
\end{enumerate}


       \subsection*{Normal families and Montel's theorem.}
       \vskip 0.5cm
       Below, $\Omega\subset\C$ will be a connected open set.
       Recall that a family $\mathcal F$ of functions $\Omega\to\C$ is
       {\it locally bounded} if every $z\in\Omega$ has a neighborhood
       $U$ and there is some $C$ such that $|f(w)|<C$ for every $w\in
       U$ and every $f\in\mathcal F$. 
       \begin{enumerate}[resume]

\item Prove the converse of Montel's theorem: if a sequence of
  holomorphic functions $f_n:\Omega\to\C$ converges uniformly on
  compact sets to a function $f:\Omega\to\C$, then the sequence is
  locally bounded.

  \item Prove Vitali's theorem: Let $\Omega\subseteq\C$ be a domain,
    let $f_n:\Omega\to\C$ be a locally
    bounded sequence of
    holomorphic functions, and let $f:\Omega\to\C$ be
    holomorphic. Let $$A=\{z\in\Omega\mid \lim_{n\to\infty} f_n(z)
    \mbox{ exists and } f(z)=\lim_{n\to\infty} f_n(z)\}$$ If $A$ has a
    limit point in $\Omega$, then $f_n\to f$ uniformly on compact sets.

       \end{enumerate}

\subsection*{Limits of holomorphic functions.}
\vskip 0.5cm
Here we will collect some properties of limits of sequences of
functions. Let $\Omega\subseteq\C$ be a connected open set (domain),
and let $f_n:\Omega\to\C$ be holomorphic functions converging
uniformly on compact sets to a function $f:\Omega\to\C$.


\begin{enumerate}[resume]
  \item The function $f$ is holomorphic. Hint: Goursat plus Morera,
    but make sure you verify all assumptions.
    \item $f_n'\to f'$ uniformly on compact sets, and likewise for
      higher derivatives. Hint: Cauchy.
    \item If $f_n(z)\neq 0$ for all $n$ and all $z\in\Omega$, then
      either $f(z)\neq 0$ for all $z\in\Omega$, or $f\equiv 0$. Hint:
      Rouch\'e.
      \item (Hurwitz) If all $f_n$ are injective, then either $f$ is injective
        or $f$ is constant.

 \end{enumerate}
\end{document}
