Remarks on the nonexistence of biharmonic maps

Abstract

In this short note we study a nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that $${\phi : (M, g) \to (N, h)}$$ is a biharmonic map, where (M, g) is a complete Riemannian manifold and (N, h) a Riemannian manifold with nonpositive sectional curvature, we will prove that $${\phi}$$ is a harmonic map if one of the following conditions holds: (i) $${|d\phi|}$$ is bounded in L q (M) and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 \leq q \leq \infty}$$ , $${1 < p < \infty}$$ ; or (ii) $${Vol(M) = \infty}$$ and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 < p < \infty}$$ . In addition, if N has strictly negative sectional curvature, we assume that $${rank\phi(q) \geq 2}$$ for some $${q \in M}$$ and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 < p < \infty}$$ . These results improve the related theorems due to Baird et al. (cf. Ann Golb Anal Geom 34:403–414, 2008), Nakauchi et al. (cf. Geom. Dedicata 164:263–272, 2014), Maeta (cf. Ann Glob Anal Geom 46:75–85, 2014), and Luo (cf. J Geom Anal 25:2436–2449, 2015).