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arXiv · 2007.09129

Rigidity and vanishing theorems for complete translating solitons

Ha Tuan Dung·Nguyen Thac Dung·Tran Quang Huy
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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.

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Ha Tuan Dung, Nguyen Thac Dung, Tran Quang Huy. 2020-07-17. Rigidity and vanishing theorems for complete translating solitons. https://arxiv.org/abs/2007.09129

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