The First Variation Of The Conformal Volume Functional In Space-like Hypersurfaces Of The Conformal Space Q₁ⁿ

The first variation of conformal volume functional is an important research subject in geometric analysis research field, is also a hot issue. And the research has been received the widespread attention by mathematicians at home and abroad.This paper gave the derivation and proof to the first variation of conformal volume functional, and introduced some preliminary knowledge at the start of the parts. Full text is as follows:The first chapter introduced the research background of the first variation, and the content structure.The second chapter reviewed the basic knowledge of hypersurface geometry and conformal differential geometry, and proved the fundamental theorem.The third chapter introduced the concept and the calculation problem of the con-formal metric fundamental equation, they are the foundation for the calculation of the first variation, because of these, we can get the derivation and proof to the first variation of conformal volume functional. The previous formal of proof is a little more complex, we take matrix form for observation, which simplified the form of the formula. At last, we give several common examples about willmore surface and hypersurface, and verify that lorentz space and stationary kind of surface is willmore.