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This note establishes a rigidity theorem and a vanishing theorem for translating solitons, indicating new insights into their properties.
In this short note, we focus on complete translating solitons with a bounded Lfn-norm of the second fundamental form and obtain two results. First, based on a Sobolev-type inequality and a Simons-type inequality, we establish a rigidity theorem of complete translating solitons. Second, based on the same Sobolev-type inequality and a Bochner-type inequality, a vanishing theorem regarding Lfp weighted harmonic 1-forms is proved.
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Peng et al. (2025) studied this question.
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