Uniqueness Results of Algebraic Curves and Related Topics



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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 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.



Uniqueness theorems, Second Main Theorem, Nevanlinna theory, Algebraic curves, Holomorphic curves, Gauss maps, Minimal surfaces