Analyticity and tautness of smooth Dupin hypersurfaces

We prove that a smooth immersed Euclidean hypersurface is locally Nash if it satisfies the Dupin condition wherever the principal multiplicities are locally constant. No local finiteness assumption on this regular locus, or regularity assumption on its complement, is required. The proof combines a dimension-dependent degree bound on proper Dupin pieces with a continuation argument using finite jets and resultants. An analytic Hessian criterion then implies that all squared distance functions are Morse-Bott, and hence that every compact embedded hypersurface in this class is taut. In each dimension there are only finitely many such hypersurfaces up to Nash isotopy through embedded Dupin hypersurfaces. We also obtain finiteness results for taut submanifolds in arbitrary codimension and derive topological restrictions from classical classification theorems for taut embeddings.