Some results of exponentially biharmonic maps into a non-positively curved manifold
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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 \begin{align*} &\int_Me^{\frac{p|\tau(u)|^2}{2}}|\tau(u)|^{p}dv_g<\infty~(p\geq2),~\int_M|\tau(u)|^{2}dv_g<\infty\text{ and}\\ &\int_M|du|^{2}dv_g<\infty, \end{align*} then $u$ is harmonic. When $u$ is an isometric immersion, we get that if $\int_Me^{\frac{pm^2|H|^2}{2}}|H|^{q}dv_g<\infty$ 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).
Abstract
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 \begin{align*} &\int_Me^{\frac{p|\tau(u)|^2}{2}}|\tau(u)|^{p}dv_g<\infty~(p\geq2),~\int_M|\tau(u)|^{2}dv_g<\infty\text{ and}\\ &\int_M|du|^{2}dv_g<\infty, \end{align*} then $u$ is harmonic. When $u$ is an isometric immersion, we get that if $\int_Me^{\frac{pm^2|H|^2}{2}}|H|^{q}dv_g<\infty$ 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).
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