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Ahlfors Second Fundamental Theorem of covering surfaces over the Riemann sphere are one of the major events in the history of function theory, which is a geometrical interpretation of the famous Nevanlinna's Second Fundamental Theorem in the theory of value distribution. The goal of this paper is to present a method to identify the precise value for that constant h in Ahlfors' SFT for covering surfaces over the unit sphere. This problem can be traced back to the early 1940s, when J. Dufresnoy first gave a numerical estimate of h for covering surfaces over a sphere. We will first prove the existence of extremal surfaces, and then we will give a formula to compute the precise value., Comment: 139 pages, 3 figures

In this paper, we will prove a result which is used by Guang-Yuan Zhang in another paper in which the existence of extremal surfaces for covering surfaces is proved and the sharp form of Ahlfors' Second Fundamental Theorem is given., Comment: 36

Ahlfors' theory of covering surfaces is one of the major mathematical achievement of last century. The most important part of his theory is the Second Fundamental Theorem (SFT). We are interested in the relation of errors of Ahlfors' SFT with the same boundary curve. In this paper we will prove a result which is used to establish the best bound of the constant in Ahlfors' SFT in arXiv.2307.04623. Precisely speaking, we will prove that for any surface $\Sigma\in\mathcal{F}_r(L,m)$, a new surface $\Sigma_1$ can be constructed based on it, such that $R(\Sigma_1)\ge R(\Sigma)$ and $L(\partial\Sigma_1)\le L(\partial\Sigma)$, where $R(\Sigma)$ is Ahlfors' error term and $L(\partial\Sigma)$ is the boundary length of the surface $\Sigma$, and the covering degree of $\Sigma_1$ has an upper bound independent of surfaces. Meanwhile, this conclusion suggests that the supremum of $H(\Sigma)=R(\Sigma)/L(\partial\Sigma)$ can be achieved by surfaces in the space $\mathcal{F}_r'(L,m)$., Comment: 21 pages, 2 figures

In Ahlfors' covering surface theory, it is well known that there exists a positive constant $h$ such that for any nonconstant holomorphic mapping $f:% \bar{\Delta}\to S,$ if $f(\Delta)\cap \{0,1,\infty \}=\emptyset ,$ then% A(f,\Delta)\leq hL(f,\partial \Delta),% where $\Delta $ is the disk $|z|<1$ in $\mathbb{C},$ $S$ is the unit Riemann sphere, $A(f,\Delta)$ is the area of the image of $\Delta $ and $% L(f,\partial \Delta)$ is the length of the image of $\partial \Delta $, both counting multiplicities. In this paper, we will show that the best lower bound for $h$ is the number h_{0}=\max_{\tau \in \lbrack 0,1]}[ \frac{\sqrt{1+\tau ^{2}}(\pi +\arcsin \tau)}{\mathrm{{arccot}\frac{\sqrt{1-\tau ^{2}}}{\sqrt{% 1+\tau ^{2}}}}}-\tau ] =4. \allowbreak 034 159 790 \allowbreak 51..., % and this is the exact estimation, i.e. there exists a sequence of holomorphic mappings $f_{n}:\bar{\Delta}\to S$ such that $% f_{n}(\Delta)\cap \{0,1,\infty \}=\emptyset $ and \lim_{n\to \infty}A(f_{n},\Delta)/L(f_{n},\partial \Delta)=h_{0}., Comment: 83 pages

Let $\Delta ^{2}$ be a ball in the complex vector space $\mathbb{C}^{2}$ centered at the origin, let $f:\Delta ^{2}\to \mathbb{C}^{2}$ be a holomorphic mapping$,$ with $f(0)=0$, and let $M$ be a positive integer. If the origin 0 is an isolated fixed point of the $M$ th iteration $f^{M}$ of $f,$ then one can define the number $\mathcal{O}_{M}(f,0)$ of periodic orbits of $f$ with period $M$ hidden at the fixed point 0, which has the meaning: any holomorphic mapping $g:\Delta ^{2}\to \mathbb{C}^{2}$ sufficiently close to $f$ in a neighborhood of the origin has exactly $% \mathcal{O}_{M}(f,0)$ distinct periodic orbits with period $M$ near the origin, provided that all fixed points of $g^{M}$ near the origin are all simple. It is known that $\mathcal{O}_{M}(f,0)\geq 1$ iff the linear part of $f$ at the origin has a periodic point of period $M.$ This paper will continue to study the number $\mathcal{O}_{M}(f,0)$. We are interested in the condition for the linear part of $f$ at the origin such that $\mathcal{O}_{M}(f,0)\geq 2.$ For a $2\times 2$ matrix $A$ that is arbitrarily given, the goal of this paper is to give a necessary and sufficient condition for $A$,$ $such that $% \mathcal{O}_{M}(f,0)\geq 2$ for all holomorphic mappings $f:\Delta ^{2}\to \mathbb{C}^{2}$ such that $f(0)=0,$ $Df(0)=A$ and that the origin 0 is an isolated fixed point of $f^{M}.$, Comment: 33 pages

We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H. Poincar\'{e}, for the condition under which a singularity of a holomorphic vector field is an isochronous center., Comment: 6 pages

This note presents a method to study center families of periodic orbits of complex holomorphic differential equations near singularities, based on some iteration properties of fixed point indices. As an application of this method, we will prove Needham's theorem in a more general version., Comment: 10 pages, To appear in Proc. Amer.Math.Soc

Let $\Delta ^{n}$ be the ball $|x|<1$ in the complex vector space $\mathbb{C}% ^{n}$, let $f:\Delta ^{n}\to \mathbb{C}^{n}$ be a holomorphic mapping and let $M$ be a positive integer. Assume that the origin $% 0=(0,..., 0)$ is an isolated fixed point of both $f$ and the $M$-th iteration $f^{M}$ of $f$. Then for each factor $m$ of $M,$ the origin is again an isolated fixed point of $f^{m}$ and the fixed point index $\mu_{f^{m}}(0)$ of $f^{m}$ at the origin is well defined, and so is the (local) Dold's index (see [\ref{Do}]) at the origin:% \begin{equation*} P_{M}(f,0)=\sum_{\tau \subset P(M)}(-1)^{#\tau}\mu_{f^{M:\tau}}(0), \end{equation*}% where $P(M)$ is the set of all primes dividing $M,$ the sum extends over all subsets $\tau $ of $P(M)$, $#\tau $is the cardinal number of $\tau $ and $% M:\tau =M(\prod_{p\in \tau}p)^{-1}$. $P_{M}(f,0)$ can be interpreted to be the number of periodic points of period $M$ of $f$ overlapped at the origin: any holomorphic mapping $% f_{1}:\Delta ^{n}\to \mathbb{C}^{n}$ sufficiently close to $f$ has exactly $P_{M}(f,0)$ distinct periodic points of period $M$ near the origin$%, $ provided that all the fixed points of $f_{1}^{M}$ near the origin are simple. Note that $f$ itself has no periodic point of period $M$ near the origin$.$ According to M. Shub and D. Sullivan's work [\ref{SS}], a necessary condition so that $P_{M}(f,0)\neq 0$ is that the linear part of $f$ at the origin has a periodic point of period $M.$ The goal of this paper is to prove that this condition is sufficient as well for holomorphic mappings., Comment: 26 pages