Bi-Lipschitz invariance of Newton polygons along gradient canyons

Abstract

We study bi-Lipschitz right-equivalence of holomorphic function germs $f:({\mathbb{C}}^2,0)\to (\mathbb{C},0)$ via polar arcs and gradient canyons. For a polar arc $\gamma$ we consider the Newton polygon of $f_x(X+\gamma (Y),Y)$ and define its augmentation by adjoining the point $(0,\mathrm{ord}f(\gamma (y),y)-1)$. We prove that the resulting augmented Newton polygon is constant along each gradient canyon of degree $>1$ and is invariant under bi-Lipschitz right-equivalence. Moreover, its compact edges decompose into a topological part and a Lipschitz part: the latter encodes, through simple intercept relations, the second-level Henry-Parusiński type invariants. As applications, we obtain two numerical bi-Lipschitz invariants attached to a canyon: its polar multiplicity and, via the Koike-Kuo-Păunescu curvature formula, the total asymptotic Gaussian curvature concentrated in it.

Revised and expanded version, with new results on multiplicities and curvature of gradient canyons