On Kuiper's conjecture

Cecil, Thomas; Chi, Quo-Shin; Jensen, Gary

On Kuiper's conjecture

Authors

Cecil, Thomas

Chi, Quo-Shin

Jensen, Gary

Description

We prove that any connected proper Dupin hypersurface in $\R^n$ 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 $\R^n$ that satisfies a certain finiteness condition. Hence any taut submanifold M in $\R^n$, whose tube $M_ε$ 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 $m \leq 4$ is algebraic.
43 pages