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Vanishing properties of $p$-harmonic $\ell$-forms on Riemannian manifolds

Nguyen Thac Dung|Pham Trong Tien
대한수학회 (The Korean Mathematical Society)|2018.09|대한수학회지 (Journal of the KMS)|제55권 제5호|pp. 1103-1129

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참고문헌

키워드

weighted Poincare inequalityflat normal bundlelocally conformally flat$p$-harmonic functionsweighted $p$-Laplacian
주제분류수학

초록

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.

Abstract

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.

참고문헌 (37)

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