Applications of nevanlinna theory to unicity problems, hypertranscendency and functional transcendency problems

Abstract

The objective of this thesis is to apply various versions of Nevanlinna theory to study certain unicity problems, hypertranscendency and functional transcendency problems of meromorphic functions. These problems can often be reduced to show the algebraic degeneracy of some holomorphic curves from the complex plane to a complex manifold, following two main themes presented in Chapter \ref{Cha:1} and Chapter \ref{Cha:2}.

The first main theme is to study the algebraic independency of meromorphic functions. This could be seen as the algebraic degeneracy of some holomorphic curves from the complex plane into the Cartesian product of one-dimensional complex projective space. The main tool used was the classical Nevanlinna theory introduced in Section \ref{sec:NTM}. Section \ref{sec:GBT} was devoted to applying Borel's Lemma and its generalisations to study the algebraic independency of the $2n$ entire functions $f_1, \dots, f_n, e^{f_1}, \dots, e^{f_n}$ (see Corollary \ref{cor:ex2}) which can be related to the famous Ax-Schanuel Theorem. To obtain more Ax-Schanuel type inequalities, we introduced Steinmetz's theorem on the reduction of functional equations in Section \ref{sec:STM}. We also included the generalisations of Steinmetz's theorem by Gross-Osgood and B. Q. Li and then made use of the ideas of the proof of these generalisations to obtain several Ax-Schanuel inequalities in Section \ref{sec:AxS}. In fact, we obtained several inequalities of the transcendence degree of $f_1, \dots, f_n, F(f_1), \dots, F(f_n)$ when $f_i$'s are entire functions with some growth restrictions and $F$ is a transcendental meromorphic function. We also considered the hypertranscendency of $fe^g$ where $f$ is a hypertranscendental function, and $g$ is an entire function in Section \ref{sec:Hype}. In particular, we obtained partial results (Theorem \ref{cor:Bank} and Corollary \ref{cor:Bank2}) of Bank's conjecture on the hypertranscendency of $\Gamma e^g$.

The second main theme is to consider the unicity of holomorphic curves using the higher dimensional Nevanlinna theory so that powerful techniques in hyperbolicity problem can be used. Applying J. -P. Demailly's notion of meromorphic partial projective connections and totally geodesic hypersurfaces, Y. Tiba proved a Second Main Theorem which allows us to obtain a uniqueness theorem for holomorphic curves intersecting totally geodesic hypersurfaces (see Theorem \ref{thm:5}). The number of sharing hypersurfaces needed is much smaller compared to the existing results. Secondly, we applied for the first time the method of jet differentials to study unique range set problems. Since the condition of being a unique range set allows one to construct an associated hypersurface in $\mathbb{P}^3$, the unicity problem was converted to a hyperbolicity problem. One can then construct a (log-pole) jet differentials vanishing on an ample divisor and hence apply the vanishing theorem of jet differentials to obtain Theorem \ref{thm:URSME}. Utilising the recent work of Siu and P\v{a}un-Sibony on Nevanlinna theory for more general Riemann surfaces, Theorem \ref{thm:URSME} was extended to functions meromorphic except for finitely many essential singularities in Section \ref{sec:URS} and Section \ref{sec:FDHM} (see Theorem \ref{thm:URSMEP} and Theorem \ref{thm:URSPR}).