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.
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Acknowledgements
The authors would like to express their thanks to the referee(s) for his/her useful notes. This leads to improve the presentation of this paper. In particular, they thank the referees for pointing out the reference [12]. This work was initial started during a stay of the second author (Nguyen Thac Dung) at VIASM. He would like to thank the staff there for hospitality and support. Ha Tuan Dung was funded by Vingroup JSC and supported by the PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2021.TS.010. Ha Tuan Dung was also funded by Hanoi Pedagogical University 2 Foundation for Sciences and Technology Development via grant number C.2020-SP2-07. Tran Quang Huy was funded by Vingroup JSC and supported by the Master Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2021.ThS.64.
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Dung, H.T., Dung, N.T. & Huy, T.Q. Rigidity and vanishing theorems for complete translating solitons. manuscripta math. 172, 331–352 (2023). https://doi.org/10.1007/s00229-022-01420-z
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DOI: https://doi.org/10.1007/s00229-022-01420-z