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Keywords:
weighted Poincaré inequality; $\delta $-stability; $L^{p}$ harmonic $1$-form; property $(\mathcal {P}_\rho )$
weighted Poincaré inequality; $\delta $-stability; $L^{p}$ harmonic $1$-form; property $(\mathcal {P}_\rho )$
Summary:
We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is $\delta $-stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^p$ harmonic $1$-forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.
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