Bulletin of the
Korean Mathematical Society BKMS

In this paper, we show that a complete translating soliton $\Sigma^m$ in $\mathbb R^n$ for the mean curvature flow is stable with respect to weighted volume functional if $\Sigma$ satisfies that the $L^m$ norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of $\Sigma$ and the Sobolev constant provided in Michael and Simon [12]. Under the same assumption, we also prove that under this upper bound, there is no non-trivial $f$-harmonic $1$-form of $L^2_f$ on $\Sigma$. With the additional assumption that $\Sigma$ is contained in an upper half-space with respect to the translating direction then it has only one end.

Keywords: Translating solitons, $L^m$ norm of the second fundamental form, $f$-stable, $f$-harmonic forms of $L^2_f$, ends

MSC numbers: Primary 53C42, 53A10, 53C44

Supported by: The first author was supported in part by the National Research Foundation of Korea (NRF-2020R1A2C1A01005698) and the second author was supported in part by the National Research Foundation of Korea (NRF-2021R1A4A1032418).