arXiv2026
We prove that a smooth immersed Euclidean submanifold is locally Nash if it satisfies the Dupin condition on the regular part of its unit normal bundle. This condition is equivalent to the Morse-Bott property of all squared distance functions and, for compact connected embedded submanifolds, to tautness over $\mathbb{Z}_2$. No local finiteness assumption on the regular locus, or regularity assumption on its complement, is required. We also prove that, in each fixed Euclidean space, every compact connected smooth taut submanifold can be joined to one of finitely many representatives by a Nash isotopy through taut embedded submanifolds. In each fixed intrinsic dimension, we also obtain only finitely many diffeomorphism types admitting taut embeddings in Euclidean spaces. We first treat hypersurfaces by combining a dimension-dependent degree bound on proper Dupin pieces with a continuation argument using finite jets and resultants. An analytic Hessian criterion then gives the Morse-Bott property, and small tubes extend the conclusions to arbitrary codimension. The degree bound also provides uniform semialgebraic complexity. Together with the semialgebraicity of tautness in parameter families and Nash triviality, this yields the finiteness and deformation results. Classical classification theorems give further topological restrictions.