Taut submanifolds are algebraic

TL;DR

This paper proves that all compact taut submanifolds in Euclidean space are algebraic, confirming a long-standing question and showing they are connected components of real algebraic varieties.

Contribution

It establishes that every compact taut submanifold in Euclidean space is a real algebraic variety, resolving a question posed by Kuiper in the 1980s.

Findings

All compact taut submanifolds are algebraic.

They are connected components of real irreducible algebraic varieties.

The result confirms a long-standing conjecture.

Abstract

We prove that every (compact) taut submanifold in Euclidean space is real algebraic, i.e., is a connected component of a real irreducible algebraic variety in the same ambient space. This answers affirmatively a question of Nicolaas Kuiper raised in the 1980s.