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Let (X,o) be a complex analytic germ. How to visualize it?
The conic structure theorem reads: (X,o) is homeomorphic to the cone over Link[X].
In ``most cases" this homeomorphism cannot be chosen differentiable (in whichever sense). The natural weaker question is: whether (X,o) is ``inner metrically conical" (IMC), i.e. whether (X,o) is bi-Lipschitz homeomorphic to the cone over its link. Any curve-germ is inner metrically conical. In higher dimensions the (non-)IMC verification is more complicated. We study this question for complex-analytic ICIS, giving necessary/sufficient criteria to be IMC. For surface germs this becomes an ``if and only if'' condition. So we get (explicitly) a lot of ICIS that are IMC's, and the other lot of ICIS that are not IMC's. Our criteria are of two types: via the polar locus/discriminant (in the general case) and via weights (for semi-weighted homogeneous ICIS). Joint work with L. Birbrair and R. Mendes Pereira |