Notes On The Udo Simon's Conjecture

In this paper we research Udo Simon conjecture in the case k=4.We obtain the main results as follows:Letψ:S²â†’Sⁿ be a linearly full minimal immersion with induced Gaussian curvature.If K is not constant and 1/10≤K≤1/6,then n=6,and the directrixφ₀ ofψhas at least 2 distinct ramified points. Consequently if 1/7＜K≤1/6,then K is constant 1/6.The paper is divided into three sections.In section 1,as an introduction,the historical background of the relevant problems,and the main method used in this article and the principal results.In section 2,we study minimal immersionψ:S²â†’Sⁿ by the method of harmonic sequences,get some basic formulas and basic lemmas.Firstly,In section 3 we use another method to prove the case k=2,3. Secondly,we give the main results about Udo Simon conjecture in the case k=4.