A'Campo curvature bumps and the Dirac phenomenon near a singular point

TL;DR

This paper explores A'Campo curvature bumps and the Dirac phenomenon near singular points of analytic functions, revealing how curvature concentrates in gradient canyons as parameters approach zero.

Contribution

It extends A'Campo's discovery using Newton-Puiseux infinitesimals and introduces the concept of gradient canyon to analyze curvature behavior near singularities.

Findings

Curvature bumps occur near singularities in level curves.

Gaussian curvature accumulates in gradient canyons as c approaches zero.

The study provides a detailed analysis of curvature concentration phenomena.

Abstract

The level curves of an analytic function germ almost always have bumps at unexpected points near the singularity. This profound discovery of N. A'Campo is fully explored in this paper for $f(z,w)\in \C\{z,w\}$, using the Newton-Puiseux infinitesimals and the notion of gradient canyon. Equally unexpected is the Dirac phenomenon: as $c\ra 0$, the total Gaussian curvature of $f(z,w)=c$ accumulates in the gradient canyons.