A'Campo Curvature Bumps and the Dirac Phenomenon Near A Singular Point

Laurentiu Paunescu (Sydney)

Abstract

The level curves of an analytic function germ can have bumps (maxima of Gaussian curvature) at unexpected points near the singularity. This phenomenon is fully explored for $f(z,w)\in \mathbb{C}\{z,w\}$ using the Newton-Puiseux infinitesimals and the notion of gradient canyon. Equally unexpected is the Dirac phenomenon: as $$c\to 0,$$ the total Gaussian curvature of $$f=c$$ accumulates in the minimal gradient canyons, and nowhere else. Our approach mimics the introduction of polar coordinates in Analytic Geometry.

This is joint work with S. Koike and T-C Kuo.