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In this paper, we show several vanishing type theorems for $p$-harmonic $\ell$-forms on Riemannian manifolds ($p\geq2$). First of all, we consider complete non-compact immersed submanifolds $M^n$ of ${N}^{n+m}$ with flat normal bundle, we prove that any $p$-harmonic $\ell$-form on $M$ is trivial if $N$ has pure curvature tensor and $M$ satisfies some geometric conditions. Then, we obtain a vanishing theorem on Riemannian manifolds with a weighted Poincar\'{e} inequality. Final, we investigate complete simply connected, locally conformally flat Riemannian manifolds $M$ and point out that there is no nontrivial $p$-harmonic $\ell$-form on $M$ provided that the Ricci curvature has suitable bound.
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참고문헌 (37)
참고문헌 신청
D. M. J. Calderbank / 2000 / Refined Kato inequalities and conformal weights in Riemannian geometry / J. Funct. Anal. 173(1):214~255
G. Carron / 1998 / Une suite exacte en $L^2$ -cohomologie / Duke Math. J. 95(2):343~372
G. Carron / 2002 / $L^2$ harmonic forms on non-compact Riemannian manifolds, in Surveys in analysis and operator theory (Canberra, 2001) / Proc. Centre Math. Appl. Austral. Nat. Univ. 40:49~59
S.-C. Chang / 2016 / Liouville properties for p-harmonic maps with finite q-energy / Trans. Amer. Math. Soc. 368(2):787~825
L.-C. Chang / 2010 / p-harmonic 1-forms on complete manifolds / Arch. Math. (Basel) 94(2):183~192
G. Carron / 1998 / Une suite exacte en
G. Carron / 2002 /
S.-C. Chang / 2016 / Liouville properties for p-harmonic maps with finite q-energy / Trans. Amer. Math. Soc. 368(2):787~825
L.-C. Chang / 2010 / p-harmonic 1-forms on complete manifolds / Arch. Math. (Basel) 94(2):183~192
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