Lipschitz normal embedding among superisolated singularities

TL;DR

This paper characterizes Lipschitz normal embedding (LNE) among superisolated hypersurface singularities, showing it occurs precisely when the projectivized tangent cone has only ordinary singularities, thus expanding known classes of LNE singularities.

Contribution

It establishes a complete criterion for LNE in superisolated hypersurface singularities based on the nature of their tangent cones, introducing an infinite family of new LNE examples.

Findings

LNE is equivalent to the tangent cone having only ordinary singularities.

Superisolated hypersurface singularities with this property form an infinite family.

LNE is rare among surface singularities, with known examples mainly minimal singularities.

Abstract

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. A complex analytic germ is said Lipschitz normally embedded (LNE) if its outer and inner metrics are bilipschitz equivalent. LNE seems to be fairly rare among surface singularities; the only known LNE surface germs outside the trivial case (straight cones) are the minimal singularities. In this paper, we show that a superisolated hypersurface singularity is LNE if and only if its projectivized tangent cone has only ordinary singularities. This provides an infinite family of LNE singularities which is radically different from the class of minimal singularities.