The Piching Phenomenon Of Compact Minimal Hypersurfaces Of A Unit Sphere

In this paper,we study the pinching phenomenon of S through the second fundamental form theory of compact minimal hypersurface in unit sphere Sn+1.The concrete content is in the following:· In Chapter 1,we introduce the background and significance of the research,including the development of a representative at home and abroad regarding this aspect.Based on this research background and profound discussion,by using deep-going analysis,it fully shows the main work’s necessity.· In Chapter 2,we give some basic concepts,symbol involved in this paper and the structure equations of Sn+1 and Mn.We obtain the second fundamental form tensor’s covariant derivatives and index transformation formulas from the structural equations and laplacian of the second basic form and some basic lemma that are used in later chapter.· In Chapter 3,we introduce the best estimation of the pinching constant when S is a function.We get the estimation of A-2B and ∑hijkl2.And then we prove the best estimation of the pinching constant when S is a function.· In Chapter 4,we introduce the best estimate of the pinching constant when S is a constant and prove it.· In Chapter 5,we have a further discussion about this.When we add a constraint:if f(Sf4-f32-S2)dM ≤ c ∫ S2(S-n)dM:for a constant-1<c<1/n + 2/3,about the second pinching phenomenon of S,we can obtain that there is positive constant δ(n)＝2n+3(1-nc)/3(1+c)such that when n ≤ S ≤ n + δ(n),that must have S = n.Especially,when c = 1/2,δ(n)=1/9gn + 2/3;when c = 1/n,δ(n)=2n2/3(n+1).And for a constant-1<c<1/2n,we assume the second gap is[n,2n].about the third pinching phenomenon of S.There exists positive constant δ(n)＝1-2nc/1+c such that when 2n<S<2n + δ(n),we have S ＝ 2n.And then we discuss when the constraint ∫(Sf4-f2/3-S2)dM≤c∫S2(S-n)dM is hold.