On Kuiper's conjecture

Abstract

We prove that any connected proper Dupin hypersurface in ${\mathbb{R}}^n$ is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in ${\mathbb{R}}^n$ that satisfies a certain finiteness condition. Hence any taut submanifold M in ${\mathbb{R}}^n$, whose tube $M_{\hat{I}}\mu$ satisfies this finiteness condition, is analytic algebraic and is a connected component of an irreducible algebraic set. In particular, we prove that every taut submanifold of dimension $m\le 4$ is algebraic.

43 pages