• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.1097-1108
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• 2015

In this paper, we investigate p-biharmonic maps u : (M, g) $\rightarrow$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if ${\int}_M|{\tau}(u)|^{{\alpha}+p}dv_g$ < ${\infty}$ and ${\int}_M|d(u)|^2dv_g$ < ${\infty}$, then u is harmonic, where ${\alpha}{\geq}0$ is a nonnegative constant and $p{\geq}2$. We also obtain that any weakly convex p-biharmonic hypersurfaces in space formN(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to Conjecture 2 (generalized Chen's conjecture for p-biharmonic submanifolds).

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1651-1670
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• 2016

In this paper, we investigate exponentially biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if $\int_{M}e^{\frac{p{\mid}r(u){\mid}^2}{2}{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$ ($p{\geq}2$), $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}d(u){\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, we get that if $\int_{M}e^{\frac{pm^2{\mid}H{\mid}^2}{2}}{\mid}H{\mid}^qdv_g$ < ${\infty}$ for 2 ${\leq}$ p < ${\infty}$ and 0 < q ${\leq}$ p < ${\infty}$, then u is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form N(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).