Why cigar can not be isometrically immersed into the 3-space

TL;DR

This paper proves that the cigar soliton cannot be isometrically immersed into three-dimensional space, extending the result to higher dimensions for steady Bryant solitons, thus answering a geometric embedding question.

Contribution

It establishes the non-existence of isometric immersions of the cigar soliton into R^3 and higher-dimensional steady Bryant solitons, providing a definitive negative answer.

Findings

Cigar soliton cannot be isometrically immersed into R^3.

Higher-dimensional steady Bryant solitons also cannot be isometrically immersed.

The result clarifies limitations on embedding certain Ricci solitons into Euclidean space.

Abstract

In this paper, we study the question if there is an isometric immersion of the cigar soliton into $R^3$. We show that the answer is negative. Similar result in higher dimensions is also true for steady Bryant solitons.