Remarks on the nonexistence of biharmonic maps (Q304074)

Let \(\phi: (M, g) \to (N, h)\) be a biharmonic map, where \((M, g)\) is a complete Riemannian manifold and \((N, h)\) a Riemannian manifold with nonpositive sectional curvature. Recall that a map \(\phi: (M, g) \to (N, h)\) is called \textit{biharmonic} if it is a critical point for the bienergy functional which is given by the square norm of tension field. A harmonic map is biharmonic, but the converse is not true in general. In this paper, the author investigates several conditions that a biharmonic map becomes harmonic. The author proves that \(\phi\) is harmonic 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) \(\text{vol}(M, g) = \infty\) and \(\int_M |\tau(\phi)|^p\, dv_g< \infty\) for some \(1< p< \infty\). In addition, if \(N\) has strictly negative sectional curvature, the assumptions that \(\mathrm{rank }\phi(q) \geq 2\) for some point \(q \in M\) and \(\int_M |\tau(\phi)|^p\, dv_g<\infty\) for some \(1 \leq q \leq \infty, 1 < p< \infty\) implies \(\phi\) is harmonic. These results improve the related results in [the author, J. Geom. Anal. 25, No. 4, 2436--2449 (2015; Zbl 1335.58012); \textit{S. Maeta}, Ann. Global Anal. Geom. 46, No. 1, 75--85 (2014; Zbl 1315.58011); \textit{N. Nakauchi} et al., Geom. Dedicata 169, 263--272 (2014; Zbl 1316.58012); \textit{P. Baird} et al., Ann. Global Anal. Geom. 34, No. 4, 403--414 (2008; Zbl 1158.53049)].