Finite Type On Boundary Of Pseudoconvex Domain

Let M be a smooth pseudoconvex hypersurface of Cn+1,p is a point in M,L be a fixed tangent vector field of type(1,0)near the point p.For any integer s ∈[1,n],Bloom defined three finite types:regular contact type as(M,p),commutator type ts(M,p)and Levi form type cs(M,p).Bloom conjectured that for any integer s ∈[1,n],as(M,p)=ts(M,p)=cs(M,p).D’Angelo defined the commutator type t(L,p)and Levi form type c(L,p)associated to L of type(1,0).D’Angelo conjectured that t(L,p)=c(L,p).In this dissertation,we mainly studies the relationship between as(M,p),ts(M,p)and cs(M,p).In addition,we also studied the relationship between t(L,p)and c(L,p).This thesis is divided into four chapters.In the first chapter,we mainly introduced the background of finite types on the boundary of pseudoconvex domains.First,we introduced the ? equations in several complex variables and the global regularity theorem for the solution of the ? equations.This leads to the local regularity problem.If the subelliptic estimates exists,then the local regularity problem has a solution.The finite type of the boundary point can determined whether there is a subelliptic estimate near the boundary point.That is,the locally defined finite type condition determine the geometric properties of the whole domain.Combined with the relevant research background and according to the existing research results,we show the motivation for our study.Finally,we give the main results of this dissertation.In the second chapter,we mainly give the notations and basic concepts involved in this paper.The basic concepts include the definitions of the regular contact type,the commutator type,the Levi form type,the Catlin multitype,the D’Angelo types and the finite ideal type.And we give some basic properties and applications of these finite types.In the third chapter,we mainly give proofs of the Bloom conjecture and the D’Angelo conjecture under the assumption that there are at least n-1 positive eigenvalues.In detail,(1)For the Bloom conjecture,let M be a smooth pseudoconvex hypersurface in Cn+1,and let p be a point of M whose Levi form has at least n-1 positive eigenvalues,then the regular contact type,the Levi type,and the commutator type are equal.(2)For the D’Angelo conjecture,let M be a smooth pseudoconvex hypersurface in Cn+1,and let p be a point of M whose Levi form has at least n-1 positive eigenvalues,we prove the equality of the commutator type and the Levi form type with respect to the tangent vector of type(1,0).In the fourth chapter,we mainly give the proof of the D’Angelo conjecture in C3,let M be a smooth pseudoconvex hypersurface in C3,and let p be a point of M,then we prove the equality of the commutator type and the Levi form type associated to the tangent vector of type(1,0).