Some results of $f$-biharmonic maps into a Riemannian manifold of non-positive sectional curvature
Bull. Korean Math. Soc. 2017 Vol. 54, No. 6, 2091-2106
Published online July 26, 2017
Printed November 30, 2017
Guoqing He, Jing Li, Peibiao Zhao
AnHui Normal University, Nanjing University of Science and Technology, Nanjing University of Science and Technology
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).
Keywords : $f$-biharmonic maps, $f$-biharmonic submanifolds
MSC numbers : 58E20, 53C21
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