Self-shrinker Type Submanifolds in the Euclidean Space

Abstract

In this paper, we devote to investigating a new class of submanifolds, named the self-shrinker type submanifolds. We define a functional and prove its critical point is a self-shrinker type submanifold. We also show that a compact self-shrinker type submanifold with the parallel mean curvature vector in the Euclidean Space $$\mathbb {R}^{n+p}$$ is a minimal submanifold in a hypersphere $$\mathbb {S}^{n+p-1}$$ of $$\mathbb {R}^{n+p}$$ .