Rigidity and vanishing theorems for complete translating solitons

TL;DR

This paper establishes rigidity and vanishing theorems for complete translating solitons, showing under certain integrability conditions that such solitons must be hyperspaces and possess no nontrivial weighted harmonic 1-forms.

Contribution

It generalizes previous results by proving new rigidity theorems and vanishing properties for complete translating solitons under integrability assumptions.

Findings

Translators with finite $L^q$-norm of trace-free second fundamental form are hyperspaces.

No nontrivial $L_f^p$ weighted harmonic 1-forms exist if the second fundamental form's $L^n$-norm is bounded.

Results extend and generalize earlier rigidity and vanishing theorems.

Abstract

In this paper, we prove some rigidity theorems for complete translating solitons. Assume that the $L^q$-norm of the trace-free second fundamental form is finite, for some $q\in\mathbb{R}$ and using a Sobolev inequality, we show that translator must be hyperspace. Our results can be considered as a generalization of \cite{Ma, WXZ16, Xin15}. We also investigate a vanishing property for translators which states that there are no nontrivial $L_f^p\ (p\geq2)$ weighted harmonic $1$-forms on ${M}$ if the $L^n$-norm of the second fundamental form is bounded.