A'Campo curvature bumps and the Dirac phenomenon near a singular point. Issue 3 (11th August 2015)

• Record Type: Journal Article
• Authors: Koike, Satoshi; Kuo, Tzee-Char; Paunescu, Laurentiu
• Is Part Of: Proceedings of the London Mathematical Society. Volume 111:Issue 3(2015:Sep.)
• Page Start: 717
• Page End: 748
• Publication Date: 2015-08-11
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms/pdv036
• Languages: English
• ISSNs: 0024-6115

Abstract

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 $c \rightarrow 0$, 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 .