On Holomorphic Curves Tangent to Real Hypersurfaces of Infinite Type

Joe Kamimoto

doi:10.1007/s12220-020-00567-z

Abstract

The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D’Angelo infinite type in {{mathbb {C}}}^n. In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra, which is an important concept in singularity theory. More precisely, equivalence conditions are given in the case of some model hypersurfaces.

Highlights

Let M be a (C∞ smooth) real hypersurface in Cn and let p lie on M
Kamimoto and of infinite type at p otherwise (the latter case will be denoted by 1(M, p) = ∞)
The class of finite type plays crucial roles in the study of the local regularity in the ∂ ̄ Neumann problem over pseudoconvex domains with smooth boundary ∂. It was shown by Catlin [4,5] that M = ∂ is of finite type at p if and only if a local subelliptic estimate at p holds

Summary

Introduction

Let M be a (C∞ smooth) real hypersurface in Cn and let p lie on M. We mainly consider the following question: Question 1 When does there exist a nonconstant holomorphic curve γ∞ tangent to M at p to infinite order?. In order to investigate the flatness of real hypersurfaces, we use holomorphic curves and Newton polyhedra of defining functions, which plays important roles in singularity theory (cf [1,2]). (6) There exists a holomorphic coordinate (z) at p such that p = 0 and the Newton polyhedron of a defining function for M on (z) The following theorem gives a sufficient condition for the existence of the curve γ∞ in Question 1 This condition is described by using Newton polyhedra of defining functions for real hypersurfaces. We always consider smooth functions, mappings, real hypersurfaces and complex curves as their respective germs without any mentioning. We use the words pure terms for any harmonic polynomial and mixed terms for any sum of monomials that are neither holomorphic nor anti-holomorphic

Newton Polyhedra for Real Hypersurfaces

Standard Coordinates

Proofs of Results in the Introduction

Two Dimensional Case

Higher Dimensional Case

Open Problems