Infinite type germs of smooth hypersurfaces in $\mathbb C^n$

TL;DR

This paper investigates the properties of smooth hypersurface germs in complex n-space, establishing a link between infinite D'Angelo type points and the existence of formal complex curves through those points.

Contribution

It demonstrates that infinite D'Angelo type boundary points imply the presence of formal complex curves within the hypersurface, revealing a new geometric property.

Findings

Infinite D'Angelo type points correspond to formal complex curves.

Existence of formal complex curves at boundary points with infinite type.

Provides insight into the local geometry of smooth hypersurfaces in complex spaces.

Abstract

In this paper, we discuss germs of smooth hypersurface in $\mathbb C^n$. We show that if a point on the boundary has infinite D'Angelo type, then there exists a formal complex curve in the hypersurface through that point.