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 $γ$ we consider the Newton polygon of $f_x(X+γ(Y),Y)$ and define its augmentation by adjoining the point $(0,\operatorname{ord} f(γ(y),y)-1)$. 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.