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.

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Occurred: Sep 24, 2026

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Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Lê-Ramanujam mu-constant problem for surface singularities: are mu-constant families in three variables topologically trivial? · mu-constant problem, surface case

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the problem as the manuscript frames it: for a holomorphic family $F$ on $\mathbb C^3\times\Delta$ with isolated critical points and constant Milnor number there is, after shrinking, a homeomorphism germ $\Phi(x,t)=(\phi_t(x),t)$ , jointly continuous with continuous inverse, with $\phi_t(0)=0$ and $f_t\circ\phi_t=f_0$ . Topological right-triviality implies constancy of the embedded topological type, so this answers the surface case in full. There is no Lean formalization. The proof uses semistable reduction, the threefold log minimal model program, the Fernández de Bobadilla-Pełka equimultiplicity theorem, Okuma's simultaneous resolution and Langer's orbifold Bogomolov-Miyaoka-Yau inequality; it was not refereed.