Rigidity and vanishing theorems for complete translating solitons

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 a translator must be a hyperspace. Our results can be considered as a generalization of Ma and Miquel (Manuscripta Math 162:115–132, 2020), Wang et al. (Pure Appl Math Q 12(4):603–619, 2016), Xin (Calc Var Partial Differ Equ 54:1995–2016, 2015). We also investigate a vanishing property for translators which states that there are no nontrivial \(L_f^p\ (p\ge 2)\) weighted harmonic 1-forms on M if the \(L^n\)-norm of the second fundamental form is bounded.