單位球上的次數三之代數超曲面

Abstract

在論文中，我們想要建構出次數三之代數極小超曲面M^{n}，在單位S^{n+1}上一類的特殊解。這其實也就是在找尋此方程式Σϕiϕijϕj-|∇ϕj|^{2}Δϕ = 0 mod ϕ的特殊解，其中ϕ:R^{n+2}→R是一個三次齊次不可分解多項式。對此，我們就可以用這類特殊解明確的描述在不同維度下，單位球上的極小超曲面。

In this thesis, we want to construct a certain class of algebraic minimal hypersurface M^{n} of degree three in the unit sphere S^{n+1}. The idea is to find a class of special solutions to the equation Σϕiϕijϕj-|∇ϕj|^{2}Δϕ = 0 mod ϕ, where ϕ:R^{n+2}→R is a homogeneous irreducible polynomial of degree three. Using these special solutions, we find certain minimal hypersurfaces in explicit form with various dimensions.