We prove a unicity theorem for two conformally diffeomorphic complete minimal surfaces immersed in $${\mathbb{R}^m}$$
 whose generalized Gauss maps $${f}$$
 and $${g}$$
 agree on the pre-image $${\cup_{j=1}^q f^{-1}(H_j)}$$
 for given hyperplanes $${H_j, 1\leq j \leq q}$$
 , in $${\mathbb{P}^{m-1}(\mathbb{C})}$$
 , located in general position, under the assumption that $${\bigcap _{j=1}^{k+1} f^{-1}(H_{i_j}) = \emptyset }$$
 . In the case when $${k=m-1}$$
 , the result obtained gives an improvement of the earlier result of Fujimoto (see Kodai Math J 16:335–354, 1993).

Unicity results for Gauss maps of minimal surfaces immersed in $${\mathbb{R}^{m}}$$ R m

The purpose of this article is to deal with uniqueness problem with truncated multiplicities for meromorphic mappings in several complex variables. We obtain a degeneracy theorem of meromorphic mappings without taking account of multiplicities of order > k in counting functions and a uniqueness theorem for meromorphic mappings sharing 2n + 2(n ≥ 2) hyperplanes in general position, which improve and extend some earlier work.

Uniqueness problem for meromorphic mappings in several complex variables with few hyperplanes

It is well-known that if two complex polynomials P and Q share two values without counting multiplicities, then they are the same. Such problem is called the value sharing problem. In this dissertation, we study the value sharing problem for algebraic and holomorphic curves, as well as give its applications. We first improve the previous result of Ru-Xu [XR07] on value sharing for algebraic mappings from a compact Riemann surface into the n-dimensional projective space that agree on the pre-image for given hyperplanes located in general position, by using a new auxiliary function. Second, we study the value sharing problem for holomorphic mappings from punctured compact Riemann surfaces into the n-dimensional projective space. We also work on p-adic holomorphic curves which is similar to the algebraic curves. In the last chapter, we apply our results to the study of minimal surfaces, namely, the uniqueness theorem for Gauss maps of two minimal surfaces in the n-dimensional Euclidean space. The article which contains the majority results of this thesis has been accepted by the International Journal of Mathematics (see [RU17]).

Uniqueness Results of Algebraic Curves and Related Topics

In this paper, we investigate the growth and fixed points of meromorphic solutions and their derivatives of higher-order nonhomogeneous linear differential equations with meromorphic coefficients. Our results extend the previous theorems due to Gundersen and Yang, Han and Yi. As an application of our results, we generalized some previous results that are related to Bruck’s conjecture.

/pdf/growth-and-fixed-points-of-meromorphic-solutions-of-1ya66mgpnp.pdf

Growth and fixed points of meromorphic solutions of nonhomogeneous linear differential equations

The main purpose of this paper is to investigate the multiple values and uniqueness problem of holomorphic mappings, that share some values or sets, from an open Riemann surface $$V$$
 with a harmonic exhaustion into a compact Riemann surface $$M,$$
 such as a Riemann sphere or torus

Multiple Values and Uniqueness Problem of Holomorphic Mappings in Riemann Surfaces

The purpose of this work is twofold. The first is to solve a uniqueness problem of meromorphic mappings posed by T. Cao and H. Yi in [1]. The second is to generalize several previous uniqueness theorems of meromorphic mappings ‘‘partially’’ sharing a few moving targets, which were given by Z. Chen and M. Ru [2], Z. Chen and Q. Yan [3], D. Thai and S. Quang [13]. 1. The uniqueness problem for hyperplanes In 1926, R. Nevanlinna [10] showed that if two meromorphic functions have the same inverse images for five distinct values, then these two functions must be identical. In 1975, the Nevanlinna’s result was generalized to the case of meromorphic mappings of C into PðCÞ by H. Fujimoto [6]. In fact, he obtained that for two linearly non-degenerate meromorphic mappings f and g of C into PðCÞ, if they have the same inverse images counted with multiplicities for 3nþ 2 hyperplanes in general position in PðCÞ, then f 1⁄4 g. Over the last few decades, there have been a lot of results related this problem. (see H. Fujimoto [7], S. Ji [9], M. Ru [11], Z. Chen and Q. Yan [15]) Let f be a linearly non-degenerate meromorphic mapping of C into PðCÞ. For each hyperplane H we denote by vð f ;HÞ the map of C into N0 such that vð f ;HÞðaÞ (a A C) is the intersection multiplicity of the image of f and H at f ðaÞ. Take q hyperplanes H1; . . . ;Hq in PðCÞ in general position and a positive integer l0. Consider the family FamðfHjg j1⁄41; f ; l0Þ of all linearly non-degenerate meromorphic mappings g : C ! PðCÞ satisfying the conditions: 485 2000 Mathematics Subject Classification. 32H30, 30D35.

/pdf/the-uniqueness-problem-for-meromorphic-mappings-with-je4ou63wks.pdf

The uniqueness problem for meromorphic mappings with truncated multiplicities

In the paper, we have two purposes. Firstly, we prove two theorems and two corollaries of normal families which improve and generalize some results of Pang and Zalcman [9], Zhang, Sun and Pang [13], Chang and Fang [2]. Secondly, we use the theory of normal families and differential equations to obtain a uniqueness theorem of entire function which is an improvement of Chang and Fang [1].

/pdf/normal-families-and-uniqueness-theorem-of-holomorphic-3scj4cyx3e.pdf

Feng Lü

Papers

The uniqueness problem for meromorphic mappings with truncated multiplicities

Normal families and uniqueness theorem of holomorphic functions