Rotational Self-Shrinkers in Euclidean Spaces

Year 2024, , 34 - 43, 23.04.2024

Kadri Arslan , Yılmaz Aydın , Betül Bulca Sokur

https://doi.org/10.36890/iejg.1330887

Abstract

The rotational embedded submanifold of $\mathbb{E}^{n+d}$ first studied by
N. Kuiper. The special examples of this type are generalized Beltrami
submanifolds and toroidals submanifold. The second named authour and at. all
recently have considered $3-$dimensional rotational embedded submanifolds in
$\mathbb{E}^{5}$. They gave some basic curvature properties of this type of
submaifolds. Self-similar flows emerge as a special solution to the mean
curvature flow that preserves the shape of the evolving submanifold. In
this article we consider self-similar submanifolds in Euclidean spaces. We
obtained some results related with self-shrinking rotational submanifolds in
Euclidean $5-$space $\mathbb{E}^{5}$. Moreover, we give the necessary and
sufficient conditions for these type of submanifolds to be homothetic
solitons for their mean curvature flows.

Keywords

Rotational submanifold, mean curvature flow, homothetic soliton, self-shrinkers

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