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arXiv · 2610.04679

Analyticity and tautness of smooth Dupin hypersurfaces

Stephan Wiesendorf
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Abstract

We consider smooth immersed Euclidean hypersurfaces satisfying the Dupin condition wherever the principal multiplicities are locally constant. We prove that in this case, every local image branch is real-analytic and locally Nash, i.e. real-analytic and semialgebraic, without any local finiteness assumption on the regular locus or any regularity assumption on the singular locus. The proof combines a dimension-dependent algebraic degree bound on proper Dupin pieces with a continuation argument using finite jets and resultants. In addition, an analytic Hessian criterion then shows that every squared distance function is Morse-Bott. The author's earlier characterization of tautness consequently implies that every compact embedded hypersurface satisfying this condition is taut.

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Stephan Wiesendorf. 2026-10-03. Analyticity and tautness of smooth Dupin hypersurfaces. https://arxiv.org/abs/2610.04679

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