The Eigenvalue And Rigidity Problem Of Submanifolds

In this paper,we introduction the spectral characterization and rigidity problem of submani-folds,which constituted three chapters.In chapter one,we introduction the background,history and the basic concept of submanifolds, self-shrinkers,and biminimal submanifolds.In chapter two,we introduction a characterization of Veronese surface and proof the following theorem:Let M is an n(n≥2)dismensional closed Willmore submanifold in an n+p dismensional unit sphere sn+p,and let λ1is the first eigenvalue of Ln=-△-n/2·3/2ρ2.Then(i)If M is tatally umbilic,then λ10.(ii)Or λ1≤n2/2.On the other hand,if λ1≥-n2/2,then either λ1=0and M is tatally umbilic；or λ1=-n2/2. In the latter case,n=2,p=2,M is Veronese surface.In chapter three,we introduction the rigidity theorem of section curvature of Willmore subm-naifolds and proof the following theorem:Let Mn is an n(n≥2)dimensional compact Willmore submanifolds in the unit sphere Sn+p(1).If K,Ⅱ,p is satisy. then Mn is totally umbilical,or Mn is Willmore tori W1,n1-S1((?)×Sn-1((?),or Mn is standard immersed of two sphere product,or Mn is Veronese sueface in S4.