Bi-Lipschitz geometry of weighted homogeneous surface singularities

We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent. L. Birbrair was supported under CNPq grant no. 300985/93-2. A. Fernandes was supported under CNPq grant no. 300393/2005-9. W. D. Neumann was supported under NSA grant H98230-06-1-011 and NSF grant no. DMS-0206464.