Variational characterizations of $\xi$-submanifolds in the Eulicdean space $\bbr^{m+p}$

TL;DR

This paper characterizes $\xi$-submanifolds in Euclidean space, extending concepts like self-shrinkers and $\lambda$-hypersurfaces, by establishing conditions, volume functionals, and stability properties, including a uniqueness result for certain stable submanifolds.

Contribution

It introduces new characterizations of $\xi$-submanifolds via modified mean curvature and volume functionals, and analyzes their stability properties, including a uniqueness theorem for stable, properly immersed submanifolds.

Findings

$\xi$-submanifolds are characterized by parallel modified mean curvature in Gaussian space.

Two weighted volume functionals $V_\xi$ and $ar V_\xi$ are introduced, with $\xi$-submanifolds as their critical points.

$m$-planes are the only properly immersed, complete $W$-stable $\xi$-submanifolds with flat normal bundle under a technical condition.

Abstract

$\xi$-submanifold in the Euclidean space $\bbr^{m+p}$ is a natural extension of the concept of self-shrinker to the mean curvature flow in $\bbr^{m+p}$. It is also a generalization of the $\lambda$-hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for $\xi$-submanifolds are established. First, it is shown that a submanifold in $\bbr^{m+p}$ is a $\xi$-submanifold if and only if its modified mean curvature is parallel when viewed as a submanifold in the Gaussian space $(\bbr^{m+p},e^{-\fr{|x|^2}{m}}\lagl\cdot,\cdot\ragl)$; Then, two weighted volume functionals $V_\xi$ and $\bar V_\xi$ are introduced and it is proved that $\xi$-submanifolds can be characterized as the critical points of these two functionals; Also, the corresponding second variation formulas are computed and the ($W$-)stability properties for $\xi$-submanifolds are systematically studied. In particular, it is proved that $m$-planes are the only properly immersed, complete $W$-stable $\xi$-submanifolds with flat normal bundle under a technical condition. It would be interesting if this additional restriction could be removed.