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Some results of f-biharmonic maps into a Riemannian manifold of non-positive sectional curvature

Jing Li|Peibiao Zhao|Guoqing He
대한수학회 (The Korean Mathematical Society)|2017.11|대한수학회보 (Bulletin of the KMS)|제54권 제6호|pp. 2091-2106

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참고문헌

키워드

$f$-biharmonic submanifolds$f$-biharmonic maps
주제분류수학

초록

The authors investigate $f$-biharmonic maps $u: (M,g)\rightarrow(N,h)$ from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature, and derive that if $\int_{M}f^{p}|\tau(u)|^{p}dv_{g}<\infty$, $\int_{M}|\tau(u)|^{2}dv_{g}<\infty$ and $\int_{M}|du|^{2}dv_{g}<\infty$, then $u$ is harmonic. When $u$ is an isometric immersion, the authors also get that if $u$ satisfies some integral conditions, then it is minimal. These results give an affirmative partial answer to conjecture 4 (generalized Chen's conjecture for $f$-biharmonic submanifolds).

Abstract

The authors investigate $f$-biharmonic maps $u: (M,g)\rightarrow(N,h)$ from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature, and derive that if $\int_{M}f^{p}|\tau(u)|^{p}dv_{g}<\infty$, $\int_{M}|\tau(u)|^{2}dv_{g}<\infty$ and $\int_{M}|du|^{2}dv_{g}<\infty$, then $u$ is harmonic. When $u$ is an isometric immersion, the authors also get that if $u$ satisfies some integral conditions, then it is minimal. These results give an affirmative partial answer to conjecture 4 (generalized Chen's conjecture for $f$-biharmonic submanifolds).

참고문헌 (33)

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