Two non existence results for the self-similar equation in Euclidean 3-space

TL;DR

This paper classifies self-similar surfaces in Euclidean 3-space, showing that only certain known types exist under specific geometric conditions, thus advancing understanding of self-similar solutions in geometric analysis.

Contribution

It establishes two non-existence results for self-similar surfaces in Euclidean 3-space, identifying the only possible configurations under foliation by circles and ruled surface conditions.

Findings

Self-similar surfaces foliated by circles are only the known surfaces of revolution.

The only ruled self-similar surfaces are cylinders over planar self-similar curves.

No other self-similar surfaces exist under these geometric constraints.

Abstract

We prove that the only self-similar surfaces of Euclidean 3-space which are foliated by circles are the self-similar surfaces of revolution discovered by S. Angenent and that the only ruled, self-similar surfaces are the cylinders over planar self-similar curves.