Loubeau, Eric; Oniciuc, Cezar Biharmonic surfaces of constant mean curvature. (English) Zbl 1322.53060 Pac. J. Math. 271, No. 1, 213-230 (2014). A map \(\phi:(M^m, g)\to (N^n,h)\) between Riemannian manifolds is a biharmonic map if it is a critical point of the bienergy functional \(E_2(\phi)=\frac{1}{2}\int_M|\tau(\phi)|^2dv_g\), where \(\tau(\phi)\) is the tension field of the map \(\phi\). Equivalently, \(\phi\) is a biharmonic map if and only if its bitension field \[ \tau_2(\phi):=-\Delta \tau(\phi)-\text{trace}_g\, \text{R}^N (\text{d}\phi(\cdot), \tau(\phi)) \text{d}\phi(\cdot)=0 \] vanishes identically. The biharmonic stress energy tensor of the map \(\phi:(M^m, g)\to (N^n,h)\) is defined to be \[ S_2(X,Y)=[\frac{1}{2}|\tau(\phi)|^2+\langle \text{d}\phi, \nabla\tau(\phi) \rangle] g(X, Y) -\langle \text{d}\phi(X), \nabla_Y\tau(\phi) \rangle-\langle \text{d}\phi(Y), \nabla_X\tau(\phi) \rangle. \] An interesting link between the bitension field and the biharmonic stress energy tensor was given by G. Jiang in [Acta Math. Sin. 30, No. 2, 220–225 (1987; Zbl 0631.58007)] as: \(\operatorname{div} S_2=-\langle\text{d} \phi, \tau_2(\phi)\rangle\). In the paper under review, the authors compute the rough Laplacian of the biharmonic stress energy tensor and use it to proved some rigidity results for biharmonic constant mean curvature surfaces. Reviewer: Ye-Lin Ou (Commerce) Cited in 9 Documents MSC: 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C43 Differential geometric aspects of harmonic maps 58E20 Harmonic maps, etc. Keywords:biharmonic maps; constant mean curvature; stress-energy tensor Citations:Zbl 0631.58007 PDFBibTeX XMLCite \textit{E. Loubeau} and \textit{C. Oniciuc}, Pac. J. Math. 271, No. 1, 213--230 (2014; Zbl 1322.53060) Full Text: DOI arXiv