Concentration of curvature and Lipschitz invariants of holomorphic functions of two variables

Abstract

By combining analytic and geometric viewpoints on the concentration of the curvature of the Milnor fibre, we prove that Lipschitz homeomorphisms preserve the zones of multi-scale curvature concentration as well as the gradient canyon structure of holomorphic functions of two variables. This yields the first new Lipschitz invariants after those discovered by Henry and Parusiński in 2003. , 32S55, 58K20 (primary), 32S05, 32S15 (secondary).The authors acknowledge the support of the Labex CEMPI grant (ANR-11-LABX-0007-01). They thank the CIRM at Luminy and the MFO at Oberwolfach for hosting them in 'Research in Pairs' programs during the preparation of this manuscript. LAURENŢ IU PȂUNESCU AND MIHAI TIBȂRLet us point out that the gradient canyons and their degrees are not topological invariants, see Example 1.2.While analytic maps do not preserve polars, we prove the analytic invariance of the canyons as a preamble for the definition of our new bi-Lipschitz invariants, Theorem 3.13: If f = g • ϕ with ϕ analytic bi-Lipschitz, then ϕ transforms canyons into canyons by preserving their degrees and multiplicities. It follows that the map ϕ establishes a bijection between the canyons of f and those of g such that the degrees d gr (γ * ) and the multiplicities mult(GC(γ * )) are the same.When we drop the analyticity assumption of the bi-Lipschitz map ϕ, the perspectives are challenging since not only that polar curves are not sent to polar curves, but we cannot prove anymore that gradient canyons are sent to gradient canyons. Up to now, the only result in full generality has been obtained by Henry and Parusiński [2, 3], namely the authors have found that the leading coefficient in the expansion (21), modulo an equivalence relation, is a bi-Lipschitz invariant. More than that, Henry and Parusiński showed in [2, 3] that a certain zone in the Milnor fibre, which is characterized by the higher order of the change of the gradient, is preserved by bi-Lipschitz homeomorphisms.Our new bi-Lipschitz invariants extend in a certain sense the discrete set of topological invariants of plane curves, but they refer to the branches of the polar curve instead of the branches of the curve {f = 0}. Our clustering description of the polar curves and their associated zones refines in a multi-scale manner the Henry-Parusiński zone. As of comparing our invariants to the Henry-Parusiński Lipschitz continuous invariants, Example 1.2 shows that they are complementary.We establish in § 4 a faithful correspondence between the concentration of curvature invariants coming from Garcia Barroso and Teissier's geometric study [1] and those coming from the Koike, Kuo and Pȃunescu analytic study [5], in particular we prove (Theorem 4.1): the contact degree d γ(τ ) and the gradient canyon GC(γ(τ )) do not depend of the direction τ of the polar γ(τ ), for generic τ . This result also contributes to the proof of our main results Theorems 5.8 and 5.9 in § 5, of which we give a brief account in the following.Let f = g • ϕ with ϕ a bi-Lipschitz homeomorphism. Even i...

Clustering polar curves

“…One has a disjoint union C(f ) = r k=1 C k (f ). The canyon disks have been defined in [PT,§5.]. With these notation we may state: Theorem 4.2.…”

“…It is shown in [PT,§5.3] that the "contact orders of canyons" is well-defined, and that it yields a more refined partition of each subset C k . We consider the tangential canyons only, i.e.…”