The Lê-Ramanujam mu-constant problem for surface singularities: are mu-constant families in three variables topologically trivial?

Theorem 1.1: every holomorphic one-parameter family $f_t$ of isolated hypersurface singularities in $\mathbb C^3$ with constant $\mu(f_t,0)$ is topologically right-trivial: after shrinking there is a homeomorphism germ $\Phi(x,t)=(\phi_t(x),t)$, with $\Phi$ and $\Phi^{-1}$ jointly continuous, $\phi_t(0)=0$, $\phi_0=\mathrm{id}$ and $f_t(\phi_t(x))=f_0(x)$. With Lê-Ramanujam and Timourian-King this gives $\mu$-constant topological triviality in every dimension. The key step shows that constant $\mu$ forces constancy of Wahl's logarithmic invariant $-P^2$ for non-log-canonical germs. Not shown: Whitney equisingularity (false in general), families over higher-dimensional parameter spaces, or Zariski's multiplicity question.

Theorem 1.1: every holomorphic one-parameter family $f_t$ of isolated hypersurface singularities in $\mathbb C^3$ with constant $\mu(f_t,0)$ is topologically right-trivial: after shrinking there is a homeomorphism germ $\Phi(x,t)=(\phi_t(x),t)$, with $\Phi$ and $\Phi^{-1}$ jointly continuous, $\phi_t(0)=0$, $\phi_0=\mathrm{id}$ and $f_t(\phi_t(x))=f_0(x)$. With Lê-Ramanujam and Timourian-King this gives $\mu$-constant topological triviality in every dimension. The key step shows that constant $\mu$ forces constancy of Wahl's logarithmic invariant $-P^2$ for non-log-canonical germs. Not shown: Whitney equisingularity (false in general), families over higher-dimensional parameter spaces, or Zariski's multiplicity question.